Code coverage tests

This page documents the degree to which the PARI/GP source code is tested by our public test suite, distributed with the source distribution in directory src/test/. This is measured by the gcov utility; we then process gcov output using the lcov frond-end.

We test a few variants depending on Configure flags on the pari.math.u-bordeaux.fr machine (x86_64 architecture), and agregate them in the final report:

The target is to exceed 90% coverage for all mathematical modules (given that branches depending on DEBUGLEVEL or DEBUGMEM are not covered). This script is run to produce the results below.

LCOV - code coverage report
Current view: top level - modules - algebras.c (source / functions) Hit Total Coverage
Test: PARI/GP v2.12.1 lcov report (development 25819-e703fe1174) Lines: 3005 3121 96.3 %
Date: 2020-09-18 06:10:04 Functions: 268 273 98.2 %
Legend: Lines: hit not hit

          Line data    Source code
       1             : /* Copyright (C) 2000  The PARI group.
       2             : 
       3             : This file is part of the PARI/GP package.
       4             : 
       5             : PARI/GP is free software; you can redistribute it and/or modify it under the
       6             : terms of the GNU General Public License as published by the Free Software
       7             : Foundation. It is distributed in the hope that it will be useful, but WITHOUT
       8             : ANY WARRANTY WHATSOEVER.
       9             : 
      10             : Check the License for details. You should have received a copy of it, along
      11             : with the package; see the file 'COPYING'. If not, write to the Free Software
      12             : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
      13             : #include "pari.h"
      14             : #include "paripriv.h"
      15             : 
      16             : #define dbg_printf(lvl) if (DEBUGLEVEL >= (lvl) + 3) err_printf
      17             : 
      18             : /********************************************************************/
      19             : /**                                                                **/
      20             : /**           ASSOCIATIVE ALGEBRAS, CENTRAL SIMPLE ALGEBRAS        **/
      21             : /**                 contributed by Aurel Page (2014)               **/
      22             : /**                                                                **/
      23             : /********************************************************************/
      24             : static GEN alg_subalg(GEN al, GEN basis);
      25             : static GEN alg_maximal_primes(GEN al, GEN P);
      26             : static GEN algnatmultable(GEN al, long D);
      27             : static GEN _tablemul_ej(GEN mt, GEN x, long j);
      28             : static GEN _tablemul_ej_Fp(GEN mt, GEN x, long j, GEN p);
      29             : static GEN _tablemul_ej_Fl(GEN mt, GEN x, long j, ulong p);
      30             : static ulong algtracei(GEN mt, ulong p, ulong expo, ulong modu);
      31             : static GEN alg_pmaximal(GEN al, GEN p);
      32             : static GEN alg_maximal(GEN al);
      33             : static GEN algtracematrix(GEN al);
      34             : static GEN algtableinit_i(GEN mt0, GEN p);
      35             : static GEN algbasisrightmultable(GEN al, GEN x);
      36             : static GEN algabstrace(GEN al, GEN x);
      37             : static GEN algbasismul(GEN al, GEN x, GEN y);
      38             : static GEN algbasismultable(GEN al, GEN x);
      39             : static GEN algbasismultable_Flm(GEN mt, GEN x, ulong m);
      40             : 
      41             : static int
      42      815809 : checkalg_i(GEN al)
      43             : {
      44             :   GEN mt, rnf;
      45      815809 :   if (typ(al) != t_VEC || lg(al) != 12) return 0;
      46      815613 :   mt = alg_get_multable(al);
      47      815613 :   if (typ(mt) != t_VEC || lg(mt) == 1 || typ(gel(mt,1)) != t_MAT) return 0;
      48      815592 :   rnf = alg_get_splittingfield(al);
      49      815592 :   if (isintzero(rnf) || !gequal0(alg_get_char(al))) return 1;
      50      459725 :   if (typ(gel(al,2)) != t_VEC || lg(gel(al,2)) == 1) return 0;
      51             :   /* not checkrnf_i: beware placeholder from alg_csa_table */
      52      459718 :   return typ(rnf)==t_VEC && lg(rnf)==13;
      53             : }
      54             : void
      55      815137 : checkalg(GEN al)
      56      815137 : { if (!checkalg_i(al)) pari_err_TYPE("checkalg [please apply alginit()]",al); }
      57             : 
      58             : static int
      59      180992 : checklat_i(GEN al, GEN lat)
      60             : {
      61             :   long N,i,j;
      62             :   GEN m,t,c;
      63      180992 :   if (typ(lat)!=t_VEC || lg(lat) != 3) return 0;
      64      180992 :   t = gel(lat,2);
      65      180992 :   if (typ(t) != t_INT && typ(t) != t_FRAC) return 0;
      66      180992 :   if (gsigne(t)<=0) return 0;
      67      180992 :   m = gel(lat,1);
      68      180992 :   if (typ(m) != t_MAT) return 0;
      69      180992 :   N = alg_get_absdim(al);
      70      180992 :   if (lg(m)-1 != N || lg(gel(m,1))-1 != N) return 0;
      71     1628886 :   for (i=1; i<=N; i++)
      72    13031067 :     for (j=1; j<=N; j++) {
      73    11583173 :       c = gcoeff(m,i,j);
      74    11583173 :       if (typ(c) != t_INT) return 0;
      75    11583173 :       if (j<i && signe(gcoeff(m,i,j))) return 0;
      76    11583173 :       if (i==j && !signe(gcoeff(m,i,j))) return 0;
      77             :     }
      78      180985 :   return 1;
      79             : }
      80      180992 : void checklat(GEN al, GEN lat)
      81      180992 : { if (!checklat_i(al,lat)) pari_err_TYPE("checklat [please apply alglathnf()]", lat); }
      82             : 
      83             : /**  ACCESSORS  **/
      84             : long
      85     4824101 : alg_type(GEN al)
      86             : {
      87     4824101 :   if (isintzero(alg_get_splittingfield(al)) || !gequal0(alg_get_char(al))) return al_TABLE;
      88     3578771 :   switch(typ(gmael(al,2,1))) {
      89      895678 :     case t_MAT: return al_CSA;
      90     2683072 :     case t_INT:
      91             :     case t_FRAC:
      92             :     case t_POL:
      93     2683072 :     case t_POLMOD: return al_CYCLIC;
      94          21 :     default: return al_NULL;
      95             :   }
      96             :   return -1; /*LCOV_EXCL_LINE*/
      97             : }
      98             : long
      99         203 : algtype(GEN al)
     100         203 : { return checkalg_i(al)? alg_type(al): al_NULL; }
     101             : 
     102             : /* absdim == dim for al_TABLE. */
     103             : long
     104      224420 : alg_get_dim(GEN al)
     105             : {
     106             :   long d;
     107      224420 :   switch(alg_type(al)) {
     108       10549 :     case al_TABLE: return lg(alg_get_multable(al))-1;
     109      213794 :     case al_CSA: return lg(alg_get_relmultable(al))-1;
     110          77 :     case al_CYCLIC: d = alg_get_degree(al); return d*d;
     111           0 :     default: pari_err_TYPE("alg_get_dim", al);
     112             :   }
     113             :   return -1; /*LCOV_EXCL_LINE*/
     114             : }
     115             : 
     116             : long
     117     1546701 : alg_get_absdim(GEN al)
     118             : {
     119     1546701 :   switch(alg_type(al)) {
     120      658149 :     case al_TABLE: return lg(alg_get_multable(al))-1;
     121      113162 :     case al_CSA: return alg_get_dim(al)*nf_get_degree(alg_get_center(al));
     122      775390 :     case al_CYCLIC:
     123      775390 :       return rnf_get_absdegree(alg_get_splittingfield(al))*alg_get_degree(al);
     124           0 :     default: pari_err_TYPE("alg_get_absdim", al);
     125             :   }
     126             :   return -1;/*LCOV_EXCL_LINE*/
     127             : }
     128             : 
     129             : long
     130        1715 : algdim(GEN al, long abs)
     131             : {
     132        1715 :   checkalg(al);
     133        1694 :   if (abs) return alg_get_absdim(al);
     134        1491 :   return alg_get_dim(al);
     135             : }
     136             : 
     137             : /* only cyclic */
     138             : GEN
     139       12936 : alg_get_auts(GEN al)
     140             : {
     141       12936 :   if (alg_type(al) != al_CYCLIC)
     142           0 :     pari_err_TYPE("alg_get_auts [non-cyclic algebra]", al);
     143       12936 :   return gel(al,2);
     144             : }
     145             : GEN
     146          91 : alg_get_aut(GEN al)
     147             : {
     148          91 :   if (alg_type(al) != al_CYCLIC)
     149           7 :     pari_err_TYPE("alg_get_aut [non-cyclic algebra]", al);
     150          84 :   return gel(alg_get_auts(al),1);
     151             : }
     152             : GEN
     153          21 : algaut(GEN al) { checkalg(al); return alg_get_aut(al); }
     154             : GEN
     155       12957 : alg_get_b(GEN al)
     156             : {
     157       12957 :   if (alg_type(al) != al_CYCLIC)
     158           7 :     pari_err_TYPE("alg_get_b [non-cyclic algebra]", al);
     159       12950 :   return gel(al,3);
     160             : }
     161             : GEN
     162          35 : algb(GEN al) { checkalg(al); return alg_get_b(al); }
     163             : 
     164             : /* only CSA */
     165             : GEN
     166      215831 : alg_get_relmultable(GEN al)
     167             : {
     168      215831 :   if (alg_type(al) != al_CSA)
     169           7 :     pari_err_TYPE("alg_get_relmultable [algebra not given via mult. table]", al);
     170      215824 :   return gel(al,2);
     171             : }
     172             : GEN
     173          42 : algrelmultable(GEN al) { checkalg(al); return alg_get_relmultable(al); }
     174             : GEN
     175          49 : alg_get_splittingdata(GEN al)
     176             : {
     177          49 :   if (alg_type(al) != al_CSA)
     178           7 :     pari_err_TYPE("alg_get_splittingdata [algebra not given via mult. table]",al);
     179          42 :   return gel(al,3);
     180             : }
     181             : GEN
     182          49 : algsplittingdata(GEN al) { checkalg(al); return alg_get_splittingdata(al); }
     183             : GEN
     184        4102 : alg_get_splittingbasis(GEN al)
     185             : {
     186        4102 :   if (alg_type(al) != al_CSA)
     187           0 :     pari_err_TYPE("alg_get_splittingbasis [algebra not given via mult. table]",al);
     188        4102 :   return gmael(al,3,2);
     189             : }
     190             : GEN
     191        4102 : alg_get_splittingbasisinv(GEN al)
     192             : {
     193        4102 :   if (alg_type(al) != al_CSA)
     194           0 :     pari_err_TYPE("alg_get_splittingbasisinv [algebra not given via mult. table]",al);
     195        4102 :   return gmael(al,3,3);
     196             : }
     197             : 
     198             : /* only cyclic and CSA */
     199             : GEN
     200     8091352 : alg_get_splittingfield(GEN al) { return gel(al,1); }
     201             : GEN
     202          91 : algsplittingfield(GEN al)
     203             : {
     204             :   long ta;
     205          91 :   checkalg(al);
     206          91 :   ta = alg_type(al);
     207          91 :   if (ta != al_CYCLIC && ta != al_CSA)
     208           7 :     pari_err_TYPE("alg_get_splittingfield [use alginit]",al);
     209          84 :   return alg_get_splittingfield(al);
     210             : }
     211             : long
     212     1228633 : alg_get_degree(GEN al)
     213             : {
     214             :   long ta;
     215     1228633 :   ta = alg_type(al);
     216     1228633 :   if (ta != al_CYCLIC && ta != al_CSA)
     217          21 :     pari_err_TYPE("alg_get_degree [use alginit]",al);
     218     1228612 :   return rnf_get_degree(alg_get_splittingfield(al));
     219             : }
     220             : long
     221         301 : algdegree(GEN al)
     222             : {
     223         301 :   checkalg(al);
     224         294 :   return alg_get_degree(al);
     225             : }
     226             : 
     227             : GEN
     228      294798 : alg_get_center(GEN al)
     229             : {
     230             :   long ta;
     231      294798 :   ta = alg_type(al);
     232      294798 :   if (ta != al_CSA && ta != al_CYCLIC)
     233           7 :     pari_err_TYPE("alg_get_center [use alginit]",al);
     234      294791 :   return rnf_get_nf(alg_get_splittingfield(al));
     235             : }
     236             : GEN
     237          70 : alg_get_splitpol(GEN al)
     238             : {
     239          70 :   long ta = alg_type(al);
     240          70 :   if (ta != al_CYCLIC && ta != al_CSA)
     241           0 :     pari_err_TYPE("alg_get_splitpol [use alginit]",al);
     242          70 :   return rnf_get_pol(alg_get_splittingfield(al));
     243             : }
     244             : GEN
     245       67340 : alg_get_abssplitting(GEN al)
     246             : {
     247       67340 :   long ta = alg_type(al), prec;
     248       67340 :   if (ta != al_CYCLIC && ta != al_CSA)
     249           0 :     pari_err_TYPE("alg_get_abssplitting [use alginit]",al);
     250       67340 :   prec = nf_get_prec(alg_get_center(al));
     251       67340 :   return rnf_build_nfabs(alg_get_splittingfield(al), prec);
     252             : }
     253             : GEN
     254        1134 : alg_get_hasse_i(GEN al)
     255             : {
     256        1134 :   long ta = alg_type(al);
     257        1134 :   if (ta != al_CYCLIC && ta != al_CSA)
     258           7 :     pari_err_TYPE("alg_get_hasse_i [use alginit]",al);
     259        1127 :   if (ta == al_CSA) pari_err_IMPL("computation of Hasse invariants over table CSA");
     260        1120 :   return gel(al,4);
     261             : }
     262             : GEN
     263         210 : alghassei(GEN al) { checkalg(al); return alg_get_hasse_i(al); }
     264             : GEN
     265        1883 : alg_get_hasse_f(GEN al)
     266             : {
     267        1883 :   long ta = alg_type(al);
     268        1883 :   if (ta != al_CYCLIC && ta != al_CSA)
     269           7 :     pari_err_TYPE("alg_get_hasse_f [use alginit]",al);
     270        1876 :   if (ta == al_CSA) pari_err_IMPL("computation of Hasse invariants over table CSA");
     271        1869 :   return gel(al,5);
     272             : }
     273             : GEN
     274         329 : alghassef(GEN al) { checkalg(al); return alg_get_hasse_f(al); }
     275             : 
     276             : /* all types */
     277             : GEN
     278        2695 : alg_get_basis(GEN al) { return gel(al,7); }
     279             : GEN
     280          49 : algbasis(GEN al) { checkalg(al); return alg_get_basis(al); }
     281             : GEN
     282       59689 : alg_get_invbasis(GEN al) { return gel(al,8); }
     283             : GEN
     284          49 : alginvbasis(GEN al) { checkalg(al); return alg_get_invbasis(al); }
     285             : GEN
     286     2224155 : alg_get_multable(GEN al) { return gel(al,9); }
     287             : GEN
     288         217 : algmultable(GEN al) { checkalg(al); return alg_get_multable(al); }
     289             : GEN
     290     5565650 : alg_get_char(GEN al) { return gel(al,10); }
     291             : GEN
     292          91 : algchar(GEN al) { checkalg(al); return alg_get_char(al); }
     293             : GEN
     294      236481 : alg_get_tracebasis(GEN al) { return gel(al,11); }
     295             : 
     296             : /* lattices */
     297             : GEN
     298      244314 : alglat_get_primbasis(GEN lat) { return gel(lat,1); }
     299             : GEN
     300      289905 : alglat_get_scalar(GEN lat) { return gel(lat,2); }
     301             : 
     302             : /** ADDITIONAL **/
     303             : 
     304             : /* no garbage collection */
     305             : static GEN
     306         777 : backtrackfacto(GEN y0, long n, GEN red, GEN pl, GEN nf, GEN data, int (*test)(GEN,GEN), GEN* fa, GEN N, GEN I)
     307             : {
     308             :   long b, i;
     309         777 :   ulong lim = 1UL << 17;
     310         777 :   long *v = new_chunk(n+1);
     311         777 :   pari_sp av = avma;
     312         777 :   for (b = 0;; b += (2*b)/(3*n) + 1)
     313          21 :   {
     314             :     GEN ny, y1, y2;
     315         798 :     set_avma(av);
     316        2317 :     for (i = 1; i <= n; i++) v[i] = -b;
     317         798 :     v[n]--;
     318             :     for(;;)
     319             :     {
     320         238 :       i = n;
     321        1225 :       while (i > 0)
     322        1204 :       { if (v[i] == b) v[i--] = -b; else { v[i]++; break; } }
     323        1036 :       if (i==0) break;
     324             : 
     325        1015 :       y1 = y0;
     326        3724 :       for (i = 1; i <= n; i++) y1 = nfadd(nf, y1, ZC_z_mul(gel(red,i), v[i]));
     327        1015 :       if (!nfchecksigns(nf, y1, pl)) continue;
     328             : 
     329         896 :       ny = absi_shallow(nfnorm(nf, y1));
     330         896 :       if (!signe(ny)) continue;
     331         896 :       ny = diviiexact(ny, gcdii(ny, N));
     332         896 :       if (!Z_issmooth(ny, lim)) continue;
     333             : 
     334         798 :       y2 = idealdivexact(nf, y1, idealadd(nf,y1,I));
     335         798 :       *fa = idealfactor(nf, y2);
     336         798 :       if (!data || test(data,*fa)) return y1;
     337             :     }
     338             :   }
     339             : }
     340             : 
     341             : /* if data == NULL, the test is skipped */
     342             : /* in the test, the factorization does not contain the known factors */
     343             : static GEN
     344         777 : factoredextchinesetest(GEN nf, GEN x, GEN y, GEN pl, GEN* fa, GEN data, int (*test)(GEN,GEN))
     345             : {
     346         777 :   pari_sp av = avma;
     347             :   long n,i;
     348         777 :   GEN x1, y0, y1, red, N, I, P = gel(x,1), E = gel(x,2);
     349         777 :   n = nf_get_degree(nf);
     350         777 :   x = idealchineseinit(nf, mkvec2(x,pl));
     351         777 :   x1 = gel(x,1);
     352         777 :   red = lg(x1) == 1? matid(n): gel(x1,1);
     353         777 :   y0 = idealchinese(nf, x, y);
     354             : 
     355         777 :   E = shallowcopy(E);
     356         777 :   if (!gequal0(y0))
     357        1981 :     for (i=1; i<lg(E); i++)
     358             :     {
     359        1204 :       long v = nfval(nf,y0,gel(P,i));
     360        1204 :       if (cmpsi(v, gel(E,i)) < 0) gel(E,i) = stoi(v);
     361             :     }
     362             :   /* N and I : known factors */
     363         777 :   I = factorbackprime(nf, P, E);
     364         777 :   N = idealnorm(nf,I);
     365             : 
     366         777 :   y1 = backtrackfacto(y0, n, red, pl, nf, data, test, fa, N, I);
     367             : 
     368             :   /* restore known factors */
     369        1981 :   for (i=1; i<lg(E); i++) gel(E,i) = stoi(nfval(nf,y1,gel(P,i)));
     370         777 :   *fa = famat_reduce(famat_mul_shallow(*fa, mkmat2(P, E)));
     371             : 
     372         777 :   gerepileall(av, 2, &y1, fa);
     373         777 :   return y1;
     374             : }
     375             : 
     376             : static GEN
     377         553 : factoredextchinese(GEN nf, GEN x, GEN y, GEN pl, GEN* fa)
     378         553 : { return factoredextchinesetest(nf,x,y,pl,fa,NULL,NULL); }
     379             : 
     380             : /** OPERATIONS ON ASSOCIATIVE ALGEBRAS algebras.c **/
     381             : 
     382             : /*
     383             : Convention:
     384             : (K/F,sigma,b) = sum_{i=0..n-1} u^i*K
     385             : t*u = u*sigma(t)
     386             : 
     387             : Natural basis:
     388             : 1<=i<=d*n^2
     389             : b_i = u^((i-1)/(dn))*ZKabs.((i-1)%(dn)+1)
     390             : 
     391             : Integral basis:
     392             : Basis of some order.
     393             : 
     394             : al:
     395             : 1- rnf of the cyclic splitting field of degree n over the center nf of degree d
     396             : 2- VEC of aut^i 1<=i<=n
     397             : 3- b in nf
     398             : 4- infinite hasse invariants (mod n) : VECSMALL of size r1, values only 0 or n/2 (if integral)
     399             : 5- finite hasse invariants (mod n) : VEC[VEC of primes, VECSMALL of hasse inv mod n]
     400             : 6- nf of the splitting field (absolute)
     401             : 7* dn^2*dn^2 matrix expressing the integral basis in terms of the natural basis
     402             : 8* dn^2*dn^2 matrix expressing the natural basis in terms of the integral basis
     403             : 9* VEC of dn^2 matrices giving the dn^2*dn^2 left multiplication tables of the integral basis
     404             : 10* characteristic of the base field (used only for algebras given by a multiplication table)
     405             : 11* trace of basis elements
     406             : 
     407             : If al is given by a multiplication table (al_TABLE), only the * fields are present.
     408             : */
     409             : 
     410             : /* assumes same center and same variable */
     411             : /* currently only works for coprime degrees */
     412             : GEN
     413          77 : algtensor(GEN al1, GEN al2, long maxord) {
     414          77 :   pari_sp av = avma;
     415             :   long v, k, d1, d2;
     416             :   GEN nf, P1, P2, aut1, aut2, b1, b2, C, rnf, aut, b, x1, x2, al;
     417             : 
     418          77 :   checkalg(al1);
     419          63 :   checkalg(al2);
     420          56 :   if (alg_type(al1) != al_CYCLIC  || alg_type(al2) != al_CYCLIC)
     421          14 :     pari_err_IMPL("tensor of non-cyclic algebras"); /* TODO: do it. */
     422             : 
     423          42 :   nf=alg_get_center(al1);
     424          42 :   if (!gequal(alg_get_center(al2),nf))
     425           7 :     pari_err_OP("tensor product [not the same center]", al1, al2);
     426             : 
     427          35 :   P1=alg_get_splitpol(al1); aut1=alg_get_aut(al1); b1=alg_get_b(al1);
     428          35 :   P2=alg_get_splitpol(al2); aut2=alg_get_aut(al2); b2=alg_get_b(al2);
     429          35 :   v=varn(P1);
     430             : 
     431          35 :   d1=alg_get_degree(al1);
     432          35 :   d2=alg_get_degree(al2);
     433          35 :   if (ugcd(d1,d2) != 1)
     434           7 :     pari_err_IMPL("tensor of cylic algebras of non-coprime degrees"); /* TODO */
     435             : 
     436          28 :   if (d1==1) return gcopy(al2);
     437          21 :   if (d2==1) return gcopy(al1);
     438             : 
     439          14 :   C = nfcompositum(nf, P1, P2, 3);
     440          14 :   rnf = rnfinit(nf,gel(C,1));
     441          14 :   x1 = gel(C,2);
     442          14 :   x2 = gel(C,3);
     443          14 :   k = itos(gel(C,4));
     444          14 :   aut = gadd(gsubst(aut2,v,x2),gmulsg(k,gsubst(aut1,v,x1)));
     445          14 :   b = nfmul(nf,nfpow_u(nf,b1,d2),nfpow_u(nf,b2,d1));
     446          14 :   al = alg_cyclic(rnf,aut,b,maxord);
     447          14 :   return gerepilecopy(av,al);
     448             : }
     449             : 
     450             : /* M an n x d Flm of rank d, n >= d. Initialize Mx = y solver */
     451             : static GEN
     452        4298 : Flm_invimage_init(GEN M, ulong p)
     453             : {
     454        4298 :   GEN v = Flm_indexrank(M, p), perm = gel(v,1);
     455        4298 :   GEN MM = rowpermute(M, perm); /* square invertible */
     456        4298 :   return mkvec2(Flm_inv(MM,p), perm);
     457             : }
     458             : /* assume Mx = y has a solution, v = Flm_invimage_init(M,p); return x */
     459             : static GEN
     460      243523 : Flm_invimage_pre(GEN v, GEN y, ulong p)
     461             : {
     462      243523 :   GEN inv = gel(v,1), perm = gel(v,2);
     463      243523 :   return Flm_Flc_mul(inv, vecsmallpermute(y, perm), p);
     464             : }
     465             : 
     466             : GEN
     467        5313 : algradical(GEN al)
     468             : {
     469        5313 :   pari_sp av = avma;
     470             :   GEN I, x, traces, K, MT, P, mt;
     471             :   long l,i,ni, n;
     472             :   ulong modu, expo, p;
     473        5313 :   checkalg(al);
     474        5313 :   P = alg_get_char(al);
     475        5313 :   mt = alg_get_multable(al);
     476        5313 :   n = alg_get_absdim(al);
     477        5313 :   dbg_printf(1)("algradical: char=%Ps, dim=%d\n", P, n);
     478        5313 :   traces = algtracematrix(al);
     479        5313 :   if (!signe(P))
     480             :   {
     481         567 :     dbg_printf(2)(" char 0, computing kernel...\n");
     482         567 :     K = ker(traces);
     483         567 :     dbg_printf(2)(" ...done.\n");
     484         567 :     ni = lg(K)-1; if (!ni) return gc_const(av, gen_0);
     485          70 :     return gerepileupto(av, K);
     486             :   }
     487        4746 :   dbg_printf(2)(" char>0, computing kernel...\n");
     488        4746 :   K = FpM_ker(traces, P);
     489        4746 :   dbg_printf(2)(" ...done.\n");
     490        4746 :   ni = lg(K)-1; if (!ni) return gc_const(av, gen_0);
     491        2835 :   if (abscmpiu(P,n)>0) return gerepileupto(av, K);
     492             : 
     493             :   /* tough case, p <= n. Ronyai's algorithm */
     494        2324 :   p = P[2]; l = 1;
     495        2324 :   expo = p; modu = p*p;
     496        2324 :   dbg_printf(2)(" char>0, hard case.\n");
     497        4697 :   while (modu<=(ulong)n) { l++; modu *= p; }
     498        2324 :   MT = ZMV_to_FlmV(mt, modu);
     499        2324 :   I = ZM_to_Flm(K,p); /* I_0 */
     500        6321 :   for (i=1; i<=l; i++) {/*compute I_i, expo = p^i, modu = p^(l+1) > n*/
     501             :     long j, lig,col;
     502        4298 :     GEN v = cgetg(ni+1, t_VECSMALL);
     503        4298 :     GEN invI = Flm_invimage_init(I, p);
     504        4298 :     dbg_printf(2)(" computing I_%d:\n", i);
     505        4298 :     traces = cgetg(ni+1,t_MAT);
     506       28770 :     for (j = 1; j <= ni; j++)
     507             :     {
     508       24472 :       GEN M = algbasismultable_Flm(MT, gel(I,j), modu);
     509       24472 :       uel(v,j) = algtracei(M, p,expo,modu);
     510             :     }
     511       28770 :     for (col=1; col<=ni; col++)
     512             :     {
     513       24472 :       GEN t = cgetg(n+1,t_VECSMALL); gel(traces,col) = t;
     514       24472 :       x = gel(I, col); /*col-th basis vector of I_{i-1}*/
     515      267995 :       for (lig=1; lig<=n; lig++)
     516             :       {
     517      243523 :         GEN y = _tablemul_ej_Fl(MT,x,lig,p);
     518      243523 :         GEN z = Flm_invimage_pre(invI, y, p);
     519      243523 :         uel(t,lig) = Flv_dotproduct(v, z, p);
     520             :       }
     521             :     }
     522        4298 :     dbg_printf(2)(" computing kernel...\n");
     523        4298 :     K = Flm_ker(traces, p);
     524        4298 :     dbg_printf(2)(" ...done.\n");
     525        4298 :     ni = lg(K)-1; if (!ni) return gc_const(av, gen_0);
     526        3997 :     I = Flm_mul(I,K,p);
     527        3997 :     expo *= p;
     528             :   }
     529        2023 :   return Flm_to_ZM(I);
     530             : }
     531             : 
     532             : /* compute the multiplication table of the element x, where mt is a
     533             :  * multiplication table in an arbitrary ring */
     534             : static GEN
     535         427 : Rgmultable(GEN mt, GEN x)
     536             : {
     537         427 :   long i, l = lg(x);
     538         427 :   GEN z = NULL;
     539        5796 :   for (i = 1; i < l; i++)
     540             :   {
     541        5369 :     GEN c = gel(x,i);
     542        5369 :     if (!gequal0(c))
     543             :     {
     544         644 :       GEN M = RgM_Rg_mul(gel(mt,i),c);
     545         644 :       z = z? RgM_add(z, M): M;
     546             :     }
     547             :   }
     548         427 :   return z;
     549             : }
     550             : 
     551             : static GEN
     552          49 : change_Rgmultable(GEN mt, GEN P, GEN Pi)
     553             : {
     554             :   GEN mt2;
     555          49 :   long lmt = lg(mt), i;
     556          49 :   mt2 = cgetg(lmt,t_VEC);
     557         476 :   for (i=1;i<lmt;i++) {
     558         427 :     GEN mti = Rgmultable(mt,gel(P,i));
     559         427 :     gel(mt2,i) = RgM_mul(Pi, RgM_mul(mti,P));
     560             :   }
     561          49 :   return mt2;
     562             : }
     563             : 
     564             : static GEN
     565       19964 : alg_quotient0(GEN al, GEN S, GEN Si, long nq, GEN p, long maps)
     566             : {
     567       19964 :   GEN mt = cgetg(nq+1,t_VEC), P, Pi, d;
     568             :   long i;
     569       19964 :   dbg_printf(3)("  alg_quotient0: char=%Ps, dim=%d, dim I=%d\n", p, alg_get_absdim(al), lg(S)-1);
     570       79267 :   for (i=1; i<=nq; i++) {
     571       59303 :     GEN mti = algbasismultable(al,gel(S,i));
     572       59303 :     if (signe(p)) gel(mt,i) = FpM_mul(Si, FpM_mul(mti,S,p), p);
     573        5257 :     else          gel(mt,i) = RgM_mul(Si, RgM_mul(mti,S));
     574             :   }
     575       19964 :   if (!signe(p) && !isint1(Q_denom(mt))) {
     576          35 :     dbg_printf(3)("  bad case: denominator=%Ps\n", Q_denom(mt));
     577          35 :     P = Q_remove_denom(Si,&d);
     578          35 :     P = ZM_hnf(P);
     579          35 :     P = RgM_Rg_div(P,d);
     580          35 :     Pi = RgM_inv(P);
     581          35 :     mt = change_Rgmultable(mt,P,Pi);
     582          35 :     Si = RgM_mul(P,Si);
     583          35 :     S = RgM_mul(S,Pi);
     584             :   }
     585       19964 :   al = algtableinit_i(mt,p);
     586       19964 :   if (maps) al = mkvec3(al,Si,S); /* algebra, proj, lift */
     587       19964 :   return al;
     588             : }
     589             : 
     590             : /* quotient of an algebra by a nontrivial two-sided ideal */
     591             : GEN
     592        2709 : alg_quotient(GEN al, GEN I, long maps)
     593             : {
     594        2709 :   pari_sp av = avma;
     595             :   GEN p, IS, ISi, S, Si;
     596             :   long n, ni;
     597             : 
     598        2709 :   checkalg(al);
     599        2709 :   p = alg_get_char(al);
     600        2709 :   n = alg_get_absdim(al);
     601        2709 :   ni = lg(I)-1;
     602             : 
     603             :   /* force first vector of complement to be the identity */
     604        2709 :   IS = shallowconcat(I, gcoeff(alg_get_multable(al),1,1));
     605        2709 :   if (signe(p)) {
     606        2681 :     IS = FpM_suppl(IS,p);
     607        2681 :     ISi = FpM_inv(IS,p);
     608             :   }
     609             :   else {
     610          28 :     IS = suppl(IS);
     611          28 :     ISi = RgM_inv(IS);
     612             :   }
     613        2709 :   S = vecslice(IS, ni+1, n);
     614        2709 :   Si = rowslice(ISi, ni+1, n);
     615        2709 :   return gerepilecopy(av, alg_quotient0(al, S, Si, n-ni, p, maps));
     616             : }
     617             : 
     618             : static GEN
     619       27335 : image_keep_first(GEN m, GEN p) /* assume first column is nonzero or m==0, no GC */
     620             : {
     621             :   GEN ir, icol, irow, M, c, x;
     622             :   long i;
     623       27335 :   if (gequal0(gel(m,1))) return zeromat(nbrows(m),0);
     624             : 
     625       27321 :   if (signe(p)) ir = FpM_indexrank(m,p);
     626        1498 :   else          ir = indexrank(m);
     627             : 
     628       27321 :   icol = gel(ir,2);
     629       27321 :   if (icol[1]==1) return extract0(m,icol,NULL);
     630             : 
     631           0 :   irow = gel(ir,1);
     632           0 :   M = extract0(m, irow, icol);
     633           0 :   c = extract0(gel(m,1), irow, NULL);
     634           0 :   if (signe(p)) x = FpM_FpC_invimage(M,c,p);
     635           0 :   else          x = inverseimage(M,c); /* TODO modulo a small prime */
     636             : 
     637           0 :   for (i=1; i<lg(x); i++)
     638             :   {
     639           0 :     if (!gequal0(gel(x,i)))
     640             :     {
     641           0 :       icol[i] = 1;
     642           0 :       vecsmall_sort(icol);
     643           0 :       return extract0(m,icol,NULL);
     644             :     }
     645             :   }
     646             : 
     647             :   return NULL; /* LCOV_EXCL_LINE */
     648             : }
     649             : 
     650             : /* z[1],...z[nz] central elements such that z[1]A + z[2]A + ... + z[nz]A = A
     651             :  * is a direct sum. idempotents ==> first basis element is identity */
     652             : GEN
     653        8092 : alg_centralproj(GEN al, GEN z, long maps)
     654             : {
     655        8092 :   pari_sp av = avma;
     656             :   GEN S, U, Ui, alq, p;
     657        8092 :   long i, iu, lz = lg(z);
     658             : 
     659        8092 :   checkalg(al);
     660        8092 :   if (typ(z) != t_VEC) pari_err_TYPE("alcentralproj",z);
     661        8085 :   p = alg_get_char(al);
     662        8085 :   dbg_printf(3)("  alg_centralproj: char=%Ps, dim=%d, #z=%d\n", p, alg_get_absdim(al), lz-1);
     663        8085 :   S = cgetg(lz,t_VEC); /* S[i] = Im(z_i) */
     664       25354 :   for (i=1; i<lz; i++)
     665             :   {
     666       17269 :     GEN mti = algbasismultable(al, gel(z,i));
     667       17269 :     gel(S,i) = image_keep_first(mti,p);
     668             :   }
     669        8085 :   U = shallowconcat1(S); /* U = [Im(z_1)|Im(z_2)|...|Im(z_nz)], n x n */
     670        8085 :   if (lg(U)-1 < alg_get_absdim(al)) pari_err_TYPE("alcentralproj [z[i]'s not surjective]",z);
     671        8078 :   if (signe(p)) Ui = FpM_inv(U,p);
     672         749 :   else          Ui = RgM_inv(U);
     673             :   if (!Ui) pari_err_BUG("alcentralproj"); /*LCOV_EXCL_LINE*/
     674             : 
     675        8078 :   alq = cgetg(lz,t_VEC);
     676       25333 :   for (iu=0,i=1; i<lz; i++)
     677             :   {
     678       17255 :     long nq = lg(gel(S,i))-1, ju = iu + nq;
     679       17255 :     GEN Si = rowslice(Ui, iu+1, ju);
     680       17255 :     gel(alq, i) = alg_quotient0(al,gel(S,i),Si,nq,p,maps);
     681       17255 :     iu = ju;
     682             :   }
     683        8078 :   return gerepilecopy(av, alq);
     684             : }
     685             : 
     686             : /* al is an al_TABLE */
     687             : static GEN
     688       18340 : algtablecenter(GEN al)
     689             : {
     690       18340 :   pari_sp av = avma;
     691             :   long n, i, j, k, ic;
     692             :   GEN C, cij, mt, p;
     693             : 
     694       18340 :   n = alg_get_absdim(al);
     695       18340 :   mt = alg_get_multable(al);
     696       18340 :   p = alg_get_char(al);
     697       18340 :   C = cgetg(n+1,t_MAT);
     698       89635 :   for (j=1; j<=n; j++)
     699             :   {
     700       71295 :     gel(C,j) = cgetg(n*n-n+1,t_COL);
     701       71295 :     ic = 1;
     702      586201 :     for (i=2; i<=n; i++) {
     703      514906 :       if (signe(p)) cij = FpC_sub(gmael(mt,i,j),gmael(mt,j,i),p);
     704       52318 :       else          cij = RgC_sub(gmael(mt,i,j),gmael(mt,j,i));
     705     7242858 :       for (k=1; k<=n; k++, ic++) gcoeff(C,ic,j) = gel(cij, k);
     706             :     }
     707             :   }
     708       18340 :   if (signe(p)) return gerepileupto(av, FpM_ker(C,p));
     709        1645 :   else          return gerepileupto(av, ker(C));
     710             : }
     711             : 
     712             : GEN
     713        4865 : algcenter(GEN al)
     714             : {
     715        4865 :   checkalg(al);
     716        4865 :   if (alg_type(al)==al_TABLE) return algtablecenter(al);
     717          28 :   return alg_get_center(al);
     718             : }
     719             : 
     720             : /* Only in positive characteristic. Assumes that al is semisimple. */
     721             : GEN
     722        4004 : algprimesubalg(GEN al)
     723             : {
     724        4004 :   pari_sp av = avma;
     725             :   GEN p, Z, F, K;
     726             :   long nz, i;
     727        4004 :   checkalg(al);
     728        4004 :   p = alg_get_char(al);
     729        4004 :   if (!signe(p)) pari_err_DOMAIN("algprimesubalg","characteristic","=",gen_0,p);
     730             : 
     731        3990 :   Z = algtablecenter(al);
     732        3990 :   nz = lg(Z)-1;
     733        3990 :   if (nz==1) return Z;
     734             : 
     735        2632 :   F = cgetg(nz+1, t_MAT);
     736       14063 :   for (i=1; i<=nz; i++) {
     737       11431 :     GEN zi = gel(Z,i);
     738       11431 :     gel(F,i) = FpC_sub(algpow(al,zi,p),zi,p);
     739             :   }
     740        2632 :   K = FpM_ker(F,p);
     741        2632 :   return gerepileupto(av, FpM_mul(Z,K,p));
     742             : }
     743             : 
     744             : static GEN
     745        9774 : _FpX_mul(void* D, GEN x, GEN y) { return FpX_mul(x,y,(GEN)D); }
     746             : static GEN
     747       26007 : _FpX_pow(void* D, GEN x, GEN n) { return FpX_powu(x,itos(n),(GEN)D); }
     748             : static GEN
     749       16233 : FpX_factorback(GEN fa, GEN p)
     750             : {
     751       16233 :   return gen_factorback(gel(fa,1), zv_to_ZV(gel(fa,2)), (void *)p, &_FpX_mul, &_FpX_pow);
     752             : }
     753             : 
     754             : static GEN
     755       14546 : out_decompose(GEN t, GEN Z, GEN P, GEN p)
     756             : {
     757       14546 :   GEN ali = gel(t,1), projm = gel(t,2), liftm = gel(t,3), pZ;
     758       14546 :   if (signe(p)) pZ = FpM_image(FpM_mul(projm,Z,p),p);
     759        1407 :   else          pZ = image(RgM_mul(projm,Z));
     760       14546 :   return mkvec5(ali, projm, liftm, pZ, P);
     761             : }
     762             : /* fa factorization of charpol(x) */
     763             : static GEN
     764        7315 : alg_decompose_from_facto(GEN al, GEN x, GEN fa, GEN Z, long mini)
     765             : {
     766        7315 :   long k = lgcols(fa)-1, k2 = mini? 1: k/2;
     767        7315 :   GEN v1 = rowslice(fa,1,k2);
     768        7315 :   GEN v2 = rowslice(fa,k2+1,k);
     769        7315 :   GEN alq, P, Q, p = alg_get_char(al);
     770        7315 :   dbg_printf(3)("  alg_decompose_from_facto\n");
     771        7315 :   if (signe(p)) {
     772        6594 :     P = FpX_factorback(v1, p);
     773        6594 :     Q = FpX_factorback(v2, p);
     774        6594 :     P = FpX_mul(P, FpXQ_inv(P,Q,p), p);
     775             :   }
     776             :   else {
     777         721 :     P = factorback(v1);
     778         721 :     Q = factorback(v2);
     779         721 :     P = RgX_mul(P, RgXQ_inv(P,Q));
     780             :   }
     781        7315 :   P = algpoleval(al, P, x);
     782        7315 :   if (signe(p)) Q = FpC_sub(col_ei(lg(P)-1,1), P, p);
     783         721 :   else          Q = gsub(gen_1, P);
     784        7315 :   if (gequal0(P) || gequal0(Q)) return NULL;
     785        7315 :   alq = alg_centralproj(al, mkvec2(P,Q), 1);
     786             : 
     787        7315 :   P = out_decompose(gel(alq,1), Z, P, p); if (mini) return P;
     788        7231 :   Q = out_decompose(gel(alq,2), Z, Q, p);
     789        7231 :   return mkvec2(P,Q);
     790             : }
     791             : 
     792             : static GEN
     793       11774 : random_pm1(long n)
     794             : {
     795       11774 :   GEN z = cgetg(n+1,t_VECSMALL);
     796             :   long i;
     797       51645 :   for (i = 1; i <= n; i++) z[i] = random_bits(5)%3 - 1;
     798       11774 :   return z;
     799             : }
     800             : 
     801             : static GEN alg_decompose(GEN al, GEN Z, long mini, GEN* pt_primelt);
     802             : /* Try to split al using x's charpoly. Return gen_0 if simple, NULL if failure.
     803             :  * And a splitting otherwise
     804             :  * If pt_primelt!=NULL, compute a primitive element of the center when simple */
     805             : static GEN
     806       13746 : try_fact(GEN al, GEN x, GEN zx, GEN Z, GEN Zal, long mini, GEN* pt_primelt)
     807             : {
     808       13746 :   GEN z, dec0, dec1, cp = algcharpoly(Zal,zx,0,1), fa, p = alg_get_char(al);
     809             :   long nfa, e;
     810       13746 :   dbg_printf(3)("  try_fact: zx=%Ps\n", zx);
     811       13746 :   if (signe(p)) fa = FpX_factor(cp,p);
     812        1330 :   else          fa = factor(cp);
     813       13746 :   dbg_printf(3)("  charpoly=%Ps\n", fa);
     814       13746 :   nfa = nbrows(fa);
     815       13746 :   if (nfa == 1) {
     816        6431 :     if (signe(p)) e = gel(fa,2)[1];
     817         609 :     else          e = itos(gcoeff(fa,1,2));
     818        6431 :     if (e == 1) {
     819        3689 :       if (pt_primelt != NULL) *pt_primelt = mkvec2(x, cp);
     820        3689 :       return gen_0;
     821             :     }
     822        2742 :     else return NULL;
     823             :   }
     824        7315 :   dec0 = alg_decompose_from_facto(al, x, fa, Z, mini);
     825        7315 :   if (!dec0) return NULL;
     826        7315 :   if (!mini) return dec0;
     827          84 :   dec1 = alg_decompose(gel(dec0,1), gel(dec0,4), 1, pt_primelt);
     828          84 :   z = gel(dec0,5);
     829          84 :   if (!isintzero(dec1)) {
     830          14 :     if (signe(p)) z = FpM_FpC_mul(gel(dec0,3),dec1,p);
     831           7 :     else          z = RgM_RgC_mul(gel(dec0,3),dec1);
     832             :   }
     833          84 :   return z;
     834             : }
     835             : static GEN
     836           7 : randcol(long n, GEN b)
     837             : {
     838           7 :   GEN N = addiu(shifti(b,1), 1);
     839             :   long i;
     840           7 :   GEN res =  cgetg(n+1,t_COL);
     841          63 :   for (i=1; i<=n; i++)
     842             :   {
     843          56 :     pari_sp av = avma;
     844          56 :     gel(res,i) = gerepileuptoint(av, subii(randomi(N),b));
     845             :   }
     846           7 :   return res;
     847             : }
     848             : /* Return gen_0 if already simple. mini: only returns a central idempotent
     849             :  * corresponding to one simple factor
     850             :  * if pt_primelt!=NULL, sets it to a primitive element of the center when simple */
     851             : static GEN
     852       20013 : alg_decompose(GEN al, GEN Z, long mini, GEN* pt_primelt)
     853             : {
     854             :   pari_sp av;
     855             :   GEN Zal, x, zx, rand, dec0, B, p;
     856       20013 :   long i, nz = lg(Z)-1;
     857             : 
     858       20013 :   if (nz == 1) {
     859        9009 :     if (pt_primelt != 0) *pt_primelt = mkvec2(zerocol(alg_get_dim(al)), pol_x(0));
     860        9009 :     return gen_0;
     861             :   }
     862       11004 :   p = alg_get_char(al);
     863       11004 :   dbg_printf(2)(" alg_decompose: char=%Ps, dim=%d, dim Z=%d\n", p, alg_get_absdim(al), nz);
     864       11004 :   Zal = alg_subalg(al,Z);
     865       11004 :   Z = gel(Zal,2);
     866       11004 :   Zal = gel(Zal,1);
     867       11004 :   av = avma;
     868             : 
     869       11004 :   rand = random_pm1(nz);
     870       11004 :   zx = zc_to_ZC(rand);
     871       11004 :   if (signe(p)) {
     872       10031 :     zx = FpC_red(zx,p);
     873       10031 :     x = ZM_zc_mul(Z,rand);
     874       10031 :     x = FpC_red(x,p);
     875             :   }
     876         973 :   else x = RgM_zc_mul(Z,rand);
     877       11004 :   dec0 = try_fact(al,x,zx,Z,Zal,mini,pt_primelt);
     878       11004 :   if (dec0) return dec0;
     879        2686 :   set_avma(av);
     880             : 
     881        2742 :   for (i=2; i<=nz; i++)
     882             :   {
     883        2735 :     dec0 = try_fact(al,gel(Z,i),col_ei(nz,i),Z,Zal,mini,pt_primelt);
     884        2735 :     if (dec0) return dec0;
     885          56 :     set_avma(av);
     886             :   }
     887           7 :   B = int2n(10);
     888             :   for (;;)
     889           0 :   {
     890           7 :     GEN x = randcol(nz,B), zx = ZM_ZC_mul(Z,x);
     891           7 :     dec0 = try_fact(al,x,zx,Z,Zal,mini,pt_primelt);
     892           7 :     if (dec0) return dec0;
     893           0 :     set_avma(av);
     894             :   }
     895             : }
     896             : 
     897             : static GEN
     898       16408 : alg_decompose_total(GEN al, GEN Z, long maps)
     899             : {
     900             :   GEN dec, sc, p;
     901             :   long i;
     902             : 
     903       16408 :   dec = alg_decompose(al, Z, 0, NULL);
     904       16408 :   if (isintzero(dec))
     905             :   {
     906        9177 :     if (maps) {
     907        6545 :       long n = alg_get_absdim(al);
     908        6545 :       al = mkvec3(al, matid(n), matid(n));
     909             :     }
     910        9177 :     return mkvec(al);
     911             :   }
     912        7231 :   p = alg_get_char(al); if (!signe(p)) p = NULL;
     913        7231 :   sc = cgetg(lg(dec), t_VEC);
     914       21693 :   for (i=1; i<lg(sc); i++) {
     915       14462 :     GEN D = gel(dec,i), a = gel(D,1), Za = gel(D,4);
     916       14462 :     GEN S = alg_decompose_total(a, Za, maps);
     917       14462 :     gel(sc,i) = S;
     918       14462 :     if (maps)
     919             :     {
     920       10206 :       GEN projm = gel(D,2), liftm = gel(D,3);
     921       10206 :       long j, lS = lg(S);
     922       27859 :       for (j=1; j<lS; j++)
     923             :       {
     924       17653 :         GEN Sj = gel(S,j), p2 = gel(Sj,2), l2 = gel(Sj,3);
     925       17653 :         if (p) p2 = FpM_mul(p2, projm, p);
     926          49 :         else   p2 = RgM_mul(p2, projm);
     927       17653 :         if (p) l2 = FpM_mul(liftm, l2, p);
     928          49 :         else   l2 = RgM_mul(liftm, l2);
     929       17653 :         gel(Sj,2) = p2;
     930       17653 :         gel(Sj,3) = l2;
     931             :       }
     932             :     }
     933             :   }
     934        7231 :   return shallowconcat1(sc);
     935             : }
     936             : 
     937             : static GEN
     938       11060 : alg_subalg(GEN al, GEN basis)
     939             : {
     940       11060 :   GEN invbasis, mt, p = alg_get_char(al);
     941       11060 :   long i, j, n = lg(basis)-1;
     942             : 
     943       11060 :   if (!signe(p)) p = NULL;
     944       11060 :   basis = shallowmatconcat(mkvec2(col_ei(n,1), basis));
     945       11060 :   if (p)
     946             :   {
     947       10066 :     basis = image_keep_first(basis,p);
     948       10066 :     invbasis = FpM_inv(basis,p);
     949             :   }
     950             :   else
     951             :   { /* FIXME use an integral variant of image_keep_first */
     952         994 :     basis = QM_ImQ_hnf(basis);
     953         994 :     invbasis = RgM_inv(basis);
     954             :   }
     955       11060 :   mt = cgetg(n+1,t_VEC);
     956       11060 :   gel(mt,1) = matid(n);
     957       37190 :   for (i = 2; i <= n; i++)
     958             :   {
     959       26130 :     GEN mtx = cgetg(n+1,t_MAT), x = gel(basis,i);
     960       26130 :     gel(mtx,1) = col_ei(n,i);
     961      165440 :     for (j = 2; j <= n; j++)
     962             :     {
     963      139310 :       GEN xy = algmul(al, x, gel(basis,j));
     964      139310 :       if (p) gel(mtx,j) = FpM_FpC_mul(invbasis, xy, p);
     965       28070 :       else   gel(mtx,j) = RgM_RgC_mul(invbasis, xy);
     966             :     }
     967       26130 :     gel(mt,i) = mtx;
     968             :   }
     969       11060 :   return mkvec2(algtableinit_i(mt,p), basis);
     970             : }
     971             : 
     972             : GEN
     973          63 : algsubalg(GEN al, GEN basis)
     974             : {
     975          63 :   pari_sp av = avma;
     976             :   GEN p;
     977          63 :   checkalg(al);
     978          63 :   if (typ(basis) != t_MAT) pari_err_TYPE("algsubalg",basis);
     979          56 :   p = alg_get_char(al);
     980          56 :   if (signe(p)) basis = RgM_to_FpM(basis,p);
     981          56 :   return gerepilecopy(av, alg_subalg(al,basis));
     982             : }
     983             : 
     984             : static int
     985       11802 : cmp_algebra(GEN x, GEN y)
     986             : {
     987             :   long d;
     988       11802 :   d = gel(x,1)[1] - gel(y,1)[1]; if (d) return d < 0? -1: 1;
     989       10605 :   d = gel(x,1)[2] - gel(y,1)[2]; if (d) return d < 0? -1: 1;
     990       10605 :   return cmp_universal(gel(x,2), gel(y,2));
     991             : }
     992             : 
     993             : GEN
     994        4081 : algsimpledec_ss(GEN al, long maps)
     995             : {
     996        4081 :   pari_sp av = avma;
     997             :   GEN Z, p, r, res, perm;
     998             :   long i, l, n;
     999        4081 :   checkalg(al);
    1000        4081 :   p = alg_get_char(al);
    1001        4081 :   dbg_printf(1)("algsimpledec_ss: char=%Ps, dim=%d\n", p, alg_get_absdim(al));
    1002        4081 :   if (signe(p)) Z = algprimesubalg(al);
    1003         245 :   else          Z = algtablecenter(al);
    1004             : 
    1005        4081 :   if (lg(Z) == 2) {/* dim Z = 1 */
    1006        2135 :     n = alg_get_absdim(al);
    1007        2135 :     set_avma(av);
    1008        2135 :     if (!maps) return mkveccopy(al);
    1009        2009 :     retmkvec(mkvec3(gcopy(al), matid(n), matid(n)));
    1010             :   }
    1011        1946 :   res = alg_decompose_total(al, Z, maps);
    1012        1946 :   l = lg(res); r = cgetg(l, t_VEC);
    1013       11123 :   for (i = 1; i < l; i++)
    1014             :   {
    1015        9177 :     GEN A = maps? gmael(res,i,1): gel(res,i);
    1016        9177 :     gel(r,i) = mkvec2(mkvecsmall2(alg_get_dim(A), lg(algtablecenter(A))),
    1017             :                       alg_get_multable(A));
    1018             :   }
    1019        1946 :   perm = gen_indexsort(r, (void*)cmp_algebra, &cmp_nodata);
    1020        1946 :   return gerepilecopy(av, vecpermute(res, perm));
    1021             : }
    1022             : 
    1023             : GEN
    1024         756 : algsimpledec(GEN al, long maps)
    1025             : {
    1026         756 :   pari_sp av = avma;
    1027             :   int ss;
    1028         756 :   GEN rad, dec, res, proj=NULL, lift=NULL;
    1029         756 :   rad = algradical(al);
    1030         756 :   ss = gequal0(rad);
    1031         756 :   if (!ss)
    1032             :   {
    1033          42 :     al = alg_quotient(al, rad, maps);
    1034          42 :     if (maps) {
    1035          14 :       proj = gel(al,2);
    1036          14 :       lift = gel(al,3);
    1037          14 :       al = gel(al,1);
    1038             :     }
    1039             :   }
    1040         756 :   dec = algsimpledec_ss(al, maps);
    1041         756 :   if (!ss && maps) /* update maps */
    1042             :   {
    1043          14 :     GEN p = alg_get_char(al);
    1044             :     long i;
    1045          42 :     for (i=1; i<lg(dec); i++)
    1046             :     {
    1047          28 :       if (signe(p))
    1048             :       {
    1049          14 :         gmael(dec,i,2) = FpM_mul(gmael(dec,i,2), proj, p);
    1050          14 :         gmael(dec,i,3) = FpM_mul(lift, gmael(dec,i,3), p);
    1051             :       }
    1052             :       else
    1053             :       {
    1054          14 :         gmael(dec,i,2) = RgM_mul(gmael(dec,i,2), proj);
    1055          14 :         gmael(dec,i,3) = RgM_mul(lift, gmael(dec,i,3));
    1056             :       }
    1057             :     }
    1058             :   }
    1059         756 :   res = mkvec2(rad, dec);
    1060         756 :   return gerepilecopy(av,res);
    1061             : }
    1062             : 
    1063             : static GEN alg_idempotent(GEN al, long n, long d);
    1064             : static GEN
    1065        6482 : try_split(GEN al, GEN x, long n, long d)
    1066             : {
    1067        6482 :   GEN cp, p = alg_get_char(al), fa, e, pol, exp, P, Q, U, u, mx, mte, ire;
    1068        6482 :   long nfa, i, smalldim = alg_get_absdim(al)+1, dim, smalli = 0;
    1069        6482 :   cp = algcharpoly(al,x,0,1);
    1070        6482 :   fa = FpX_factor(cp,p);
    1071        6482 :   nfa = nbrows(fa);
    1072        6482 :   if (nfa == 1) return NULL;
    1073        3052 :   pol = gel(fa,1);
    1074        3052 :   exp = gel(fa,2);
    1075             : 
    1076             :   /* charpoly is always a d-th power */
    1077        9254 :   for (i=1; i<lg(exp); i++) {
    1078        6209 :     if (exp[i]%d) pari_err(e_MISC, "the algebra must be simple (try_split 1)");
    1079        6202 :     exp[i] /= d;
    1080             :   }
    1081        3045 :   cp = FpX_factorback(fa,p);
    1082             : 
    1083             :   /* find smallest Fp-dimension of a characteristic space */
    1084        9247 :   for (i=1; i<lg(pol); i++) {
    1085        6202 :     dim = degree(gel(pol,i))*exp[i];
    1086        6202 :     if (dim < smalldim) {
    1087        3115 :       smalldim = dim;
    1088        3115 :       smalli = i;
    1089             :     }
    1090             :   }
    1091        3045 :   i = smalli;
    1092        3045 :   if (smalldim != n) return NULL;
    1093             :   /* We could also compute e*al*e and try again with this smaller algebra */
    1094             :   /* Fq-rank 1 = Fp-rank n idempotent: success */
    1095             : 
    1096             :   /* construct idempotent */
    1097        3031 :   mx = algbasismultable(al,x);
    1098        3031 :   P = gel(pol,i);
    1099        3031 :   P = FpX_powu(P, exp[i], p);
    1100        3031 :   Q = FpX_div(cp, P, p);
    1101        3031 :   e = algpoleval(al, Q, mkvec2(x,mx));
    1102        3031 :   U = FpXQ_inv(Q, P, p);
    1103        3031 :   u = algpoleval(al, U, mkvec2(x,mx));
    1104        3031 :   e = algbasismul(al, e, u);
    1105        3031 :   mte = algbasisrightmultable(al,e);
    1106        3031 :   ire = FpM_indexrank(mte,p);
    1107        3031 :   if (lg(gel(ire,1))-1 != smalldim*d) pari_err(e_MISC, "the algebra must be simple (try_split 2)");
    1108             : 
    1109        3024 :   return mkvec3(e,mte,ire);
    1110             : }
    1111             : 
    1112             : /*
    1113             :  * Given a simple algebra al of dimension d^2 over its center of degree n,
    1114             :  * find an idempotent e in al with rank n (which is minimal).
    1115             : */
    1116             : static GEN
    1117        3038 : alg_idempotent(GEN al, long n, long d)
    1118             : {
    1119        3038 :   pari_sp av = avma;
    1120        3038 :   long i, N = alg_get_absdim(al);
    1121        3038 :   GEN e, p = alg_get_char(al), x;
    1122        6377 :   for(i=2; i<=N; i++) {
    1123        6321 :     x = col_ei(N,i);
    1124        6321 :     e = try_split(al, x, n, d);
    1125        6307 :     if (e) return e;
    1126        3339 :     set_avma(av);
    1127             :   }
    1128             :   for(;;) {
    1129         161 :     x = random_FpC(N,p);
    1130         161 :     e = try_split(al, x, n, d);
    1131         161 :     if (e) return e;
    1132         105 :     set_avma(av);
    1133             :   }
    1134             : }
    1135             : 
    1136             : static GEN
    1137        3857 : try_descend(GEN M, GEN B, GEN p, long m, long n, long d)
    1138             : {
    1139        3857 :   GEN B2 = cgetg(m+1,t_MAT), b;
    1140        3857 :   long i, j, k=0;
    1141       11011 :   for (i=1; i<=d; i++)
    1142             :   {
    1143        7154 :     k++;
    1144        7154 :     b = gel(B,i);
    1145        7154 :     gel(B2,k) = b;
    1146       17248 :     for (j=1; j<n; j++)
    1147             :     {
    1148       10094 :       k++;
    1149       10094 :       b = FpM_FpC_mul(M,b,p);
    1150       10094 :       gel(B2,k) = b;
    1151             :     }
    1152             :   }
    1153        3857 :   if (!signe(FpM_det(B2,p))) return NULL;
    1154        3437 :   return FpM_inv(B2,p);
    1155             : }
    1156             : 
    1157             : /* Given an m*m matrix M with irreducible charpoly over F of degree n,
    1158             :  * let K = F(M), which is a field, and write m=d*n.
    1159             :  * Compute the d-dimensional K-vector space structure on V=F^m induced by M.
    1160             :  * Return [B,C] where:
    1161             :  *  - B is m*d matrix over F giving a K-basis b_1,...,b_d of V
    1162             :  *  - C is d*m matrix over F[x] expressing the canonical F-basis of V on the b_i
    1163             :  * Currently F = Fp TODO extend this. */
    1164             : static GEN
    1165        3437 : descend_i(GEN M, long n, GEN p)
    1166             : {
    1167             :   GEN B, C;
    1168             :   long m,d,i;
    1169             :   pari_sp av;
    1170        3437 :   m = lg(M)-1;
    1171        3437 :   d = m/n;
    1172        3437 :   B = cgetg(d+1,t_MAT);
    1173        3437 :   av = avma;
    1174             : 
    1175             :   /* try a subset of the canonical basis */
    1176        9751 :   for (i=1; i<=d; i++)
    1177        6314 :     gel(B,i) = col_ei(m,n*(i-1)+1);
    1178        3437 :   C = try_descend(M,B,p,m,n,d);
    1179        3437 :   if (C) return mkvec2(B,C);
    1180         385 :   set_avma(av);
    1181             : 
    1182             :   /* try smallish elements */
    1183        1155 :   for (i=1; i<=d; i++)
    1184         770 :     gel(B,i) = FpC_red(zc_to_ZC(random_pm1(m)),p);
    1185         385 :   C = try_descend(M,B,p,m,n,d);
    1186         385 :   if (C) return mkvec2(B,C);
    1187          35 :   set_avma(av);
    1188             : 
    1189             :   /* try random elements */
    1190             :   for (;;)
    1191             :   {
    1192         105 :     for (i=1; i<=d; i++)
    1193          70 :       gel(B,i) = random_FpC(m,p);
    1194          35 :     C = try_descend(M,B,p,m,n,d);
    1195          35 :     if (C) return mkvec2(B,C);
    1196           0 :     set_avma(av);
    1197             :   }
    1198             : }
    1199             : static GEN
    1200       15568 : RgC_contract(GEN C, long n, long v) /* n>1 */
    1201             : {
    1202             :   GEN C2, P;
    1203             :   long m, d, i, j;
    1204       15568 :   m = lg(C)-1;
    1205       15568 :   d = m/n;
    1206       15568 :   C2 = cgetg(d+1,t_COL);
    1207       43344 :   for (i=1; i<=d; i++)
    1208             :   {
    1209       27776 :     P = pol_xn(n-1,v);
    1210      105728 :     for (j=1; j<=n; j++)
    1211       77952 :       gel(P,j+1) = gel(C,n*(i-1)+j);
    1212       27776 :     P = normalizepol(P);
    1213       27776 :     gel(C2,i) = P;
    1214             :   }
    1215       15568 :   return C2;
    1216             : }
    1217             : static GEN
    1218        3437 : RgM_contract(GEN A, long n, long v) /* n>1 */
    1219             : {
    1220        3437 :   GEN A2 = cgetg(lg(A),t_MAT);
    1221             :   long i;
    1222       19005 :   for (i=1; i<lg(A2); i++)
    1223       15568 :     gel(A2,i) = RgC_contract(gel(A,i),n,v);
    1224        3437 :   return A2;
    1225             : }
    1226             : static GEN
    1227        3437 : descend(GEN M, long n, GEN p, long v)
    1228             : {
    1229        3437 :   GEN res = descend_i(M,n,p);
    1230        3437 :   gel(res,2) = RgM_contract(gel(res,2),n,v);
    1231        3437 :   return res;
    1232             : }
    1233             : 
    1234             : /* isomorphism of Fp-vector spaces M_d(F_p^n) -> (F_p)^(d^2*n) */
    1235             : static GEN
    1236       29939 : Fq_mat2col(GEN M, long d, long n)
    1237             : {
    1238       29939 :   long N = d*d*n, i, j, k;
    1239       29939 :   GEN C = cgetg(N+1, t_COL);
    1240       90160 :   for (i=1; i<=d; i++)
    1241      191632 :     for (j=1; j<=d; j++)
    1242      400526 :       for (k=0; k<n; k++)
    1243      269115 :         gel(C,n*(d*(i-1)+j-1)+k+1) = polcoef_i(gcoeff(M,i,j),k,-1);
    1244       29939 :   return C;
    1245             : }
    1246             : 
    1247             : static GEN
    1248        3752 : alg_finite_csa_split(GEN al, long v)
    1249             : {
    1250             :   GEN Z, e, mte, ire, primelt, b, T, M, proje, lifte, extre, p, B, C, mt, mx, map, mapi, T2, ro;
    1251        3752 :   long n, d, N = alg_get_absdim(al), i;
    1252        3752 :   p = alg_get_char(al);
    1253             :   /* compute the center */
    1254        3752 :   Z = algcenter(al);
    1255             :   /* TODO option to give the center as input instead of computing it */
    1256        3752 :   n = lg(Z)-1;
    1257             : 
    1258             :   /* compute a minimal rank idempotent e */
    1259        3752 :   if (n==N) {
    1260         707 :     d = 1;
    1261         707 :     e = col_ei(N,1);
    1262         707 :     mte = matid(N);
    1263         707 :     ire = mkvec2(identity_perm(n),identity_perm(n));
    1264             :   }
    1265             :   else {
    1266        3045 :     d = usqrt(N/n);
    1267        3045 :     if (d*d*n != N) pari_err(e_MISC, "the algebra must be simple (alg_finite_csa_split 1)");
    1268        3038 :     e = alg_idempotent(al,n,d);
    1269        3024 :     mte = gel(e,2);
    1270        3024 :     ire = gel(e,3);
    1271        3024 :     e = gel(e,1);
    1272             :   }
    1273             : 
    1274             :   /* identify the center */
    1275        3731 :   if (n==1)
    1276             :   {
    1277         287 :     T = pol_x(v);
    1278         287 :     primelt = gen_0;
    1279             :   }
    1280             :   else
    1281             :   {
    1282        3444 :     b = alg_decompose(al, Z, 1, &primelt);
    1283        3444 :     if (!gequal0(b)) pari_err(e_MISC, "the algebra must be simple (alg_finite_csa_split 2)");
    1284        3437 :     T = gel(primelt,2);
    1285        3437 :     primelt = gel(primelt,1);
    1286        3437 :     setvarn(T,v);
    1287             :   }
    1288             : 
    1289             :   /* use the ffinit polynomial */
    1290        3724 :   if (n>1)
    1291             :   {
    1292        3437 :     T2 = init_Fq(p,n,v);
    1293        3437 :     setvarn(T,fetch_var_higher());
    1294        3437 :     ro = FpXQX_roots(T2,T,p);
    1295        3437 :     ro = gel(ro,1);
    1296        3437 :     primelt = algpoleval(al,ro,primelt);
    1297        3437 :     T = T2;
    1298             :   }
    1299             : 
    1300             :   /* descend al*e to a vector space over the center */
    1301             :   /* lifte: al*e -> al ; proje: al*e -> al */
    1302        3724 :   lifte = shallowextract(mte,gel(ire,2));
    1303        3724 :   extre = shallowmatextract(mte,gel(ire,1),gel(ire,2));
    1304        3724 :   extre = FpM_inv(extre,p);
    1305        3724 :   proje = rowpermute(mte,gel(ire,1));
    1306        3724 :   proje = FpM_mul(extre,proje,p);
    1307        3724 :   if (n==1)
    1308             :   {
    1309         287 :     B = lifte;
    1310         287 :     C = proje;
    1311             :   }
    1312             :   else
    1313             :   {
    1314        3437 :     M = algbasismultable(al,primelt);
    1315        3437 :     M = FpM_mul(M,lifte,p);
    1316        3437 :     M = FpM_mul(proje,M,p);
    1317        3437 :     B = descend(M,n,p,v);
    1318        3437 :     C = gel(B,2);
    1319        3437 :     B = gel(B,1);
    1320        3437 :     B = FpM_mul(lifte,B,p);
    1321        3437 :     C = FqM_mul(C,proje,T,p);
    1322             :   }
    1323             : 
    1324             :   /* compute the isomorphism */
    1325        3724 :   mt = alg_get_multable(al);
    1326        3724 :   map = cgetg(N+1,t_VEC);
    1327        3724 :   M = cgetg(N+1,t_MAT);
    1328       33663 :   for (i=1; i<=N; i++)
    1329             :   {
    1330       29939 :     mx = gel(mt,i);
    1331       29939 :     mx = FpM_mul(mx,B,p);
    1332       29939 :     mx = FqM_mul(C,mx,T,p);
    1333       29939 :     gel(map,i) = mx;
    1334       29939 :     gel(M,i) = Fq_mat2col(mx,d,n);
    1335             :   }
    1336        3724 :   mapi = FpM_inv(M,p);
    1337        3724 :   if (!mapi) pari_err(e_MISC, "the algebra must be simple (alg_finite_csa_split 3)");
    1338        3717 :   return mkvec3(T,map,mapi);
    1339             : }
    1340             : 
    1341             : GEN
    1342        3766 : algsplit(GEN al, long v)
    1343             : {
    1344        3766 :   pari_sp av = avma;
    1345             :   GEN res, T, map, mapi, ff, p;
    1346             :   long i,j,k,li,lj;
    1347        3766 :   checkalg(al);
    1348        3759 :   p = alg_get_char(al);
    1349        3759 :   if (gequal0(p))
    1350           7 :     pari_err_IMPL("splitting a characteristic 0 algebra over its center");
    1351        3752 :   res = alg_finite_csa_split(al, v);
    1352        3717 :   T = gel(res,1);
    1353        3717 :   map = gel(res,2);
    1354        3717 :   mapi = gel(res,3);
    1355        3717 :   ff = Tp_to_FF(T,p);
    1356       33593 :   for (i=1; i<lg(map); i++)
    1357             :   {
    1358       29876 :     li = lg(gel(map,i));
    1359       89908 :     for (j=1; j<li; j++)
    1360             :     {
    1361       60032 :       lj = lg(gmael(map,i,j));
    1362      190876 :       for (k=1; k<lj; k++)
    1363      130844 :         gmael3(map,i,j,k) = Fq_to_FF(gmael3(map,i,j,k),ff);
    1364             :     }
    1365             :   }
    1366             : 
    1367        3717 :   return gerepilecopy(av, mkvec2(map,mapi));
    1368             : }
    1369             : 
    1370             : /* multiplication table sanity checks */
    1371             : static GEN
    1372       36841 : check_mt_noid(GEN mt, GEN p)
    1373             : {
    1374             :   long i, l;
    1375       36841 :   GEN MT = cgetg_copy(mt, &l);
    1376       36841 :   if (typ(MT) != t_VEC || l == 1) return NULL;
    1377      179093 :   for (i = 1; i < l; i++)
    1378             :   {
    1379      142294 :     GEN M = gel(mt,i);
    1380      142294 :     if (typ(M) != t_MAT || lg(M) != l || lgcols(M) != l) return NULL;
    1381      142273 :     if (p) M = RgM_to_FpM(M,p);
    1382      142273 :     gel(MT,i) = M;
    1383             :   }
    1384       36799 :   return MT;
    1385             : }
    1386             : static GEN
    1387       36379 : check_mt(GEN mt, GEN p)
    1388             : {
    1389             :   long i;
    1390             :   GEN MT;
    1391       36379 :   MT = check_mt_noid(mt, p);
    1392       36379 :   if (!MT || !ZM_isidentity(gel(MT,1))) return NULL;
    1393      139228 :   for (i=2; i<lg(MT); i++)
    1394      102870 :     if (ZC_is_ei(gmael(MT,i,1)) != i) return NULL;
    1395       36358 :   return MT;
    1396             : }
    1397             : 
    1398             : static GEN
    1399         161 : check_relmt(GEN nf, GEN mt)
    1400             : {
    1401         161 :   long i, l = lg(mt), j, k;
    1402         161 :   GEN MT = gcopy(mt), a, b, d;
    1403         161 :   if (typ(MT) != t_VEC || l == 1) return NULL;
    1404         623 :   for (i = 1; i < l; i++)
    1405             :   {
    1406         483 :     GEN M = gel(MT,i);
    1407         483 :     if (typ(M) != t_MAT || lg(M) != l || lgcols(M) != l) return NULL;
    1408        2478 :     for (k = 1; k < l; k++)
    1409       12523 :       for (j = 1; j < l; j++)
    1410             :       {
    1411       10528 :         a = gcoeff(M,j,k);
    1412       10528 :         if (typ(a)==t_INT) continue;
    1413        1771 :         b = algtobasis(nf,a);
    1414        1771 :         d = Q_denom(b);
    1415        1771 :         if (!isint1(d))
    1416          14 :           pari_err_DOMAIN("alg_csa_table", "denominator(mt)", "!=", gen_1, mt);
    1417        1757 :         gcoeff(M,j,k) = lift(basistoalg(nf,b));
    1418             :       }
    1419         469 :     if (i > 1 && RgC_is_ei(gel(M,1)) != i) return NULL; /* i = 1 checked at end */
    1420         462 :     gel(MT,i) = M;
    1421             :   }
    1422         140 :   if (!RgM_isidentity(gel(MT,1))) return NULL;
    1423         140 :   return MT;
    1424             : }
    1425             : 
    1426             : int
    1427         469 : algisassociative(GEN mt0, GEN p)
    1428             : {
    1429         469 :   pari_sp av = avma;
    1430             :   long i, j, k, n;
    1431             :   GEN M, mt;
    1432             : 
    1433         469 :   if (checkalg_i(mt0)) { p = alg_get_char(mt0); mt0 = alg_get_multable(mt0); }
    1434         469 :   if (typ(p) != t_INT) pari_err_TYPE("algisassociative",p);
    1435         462 :   mt = check_mt_noid(mt0, isintzero(p)? NULL: p);
    1436         462 :   if (!mt) pari_err_TYPE("algisassociative (mult. table)", mt0);
    1437         427 :   if (!ZM_isidentity(gel(mt,1))) return gc_bool(av,0);
    1438         413 :   n = lg(mt)-1;
    1439         413 :   M = cgetg(n+1,t_MAT);
    1440        3402 :   for (j=1; j<=n; j++) gel(M,j) = cgetg(n+1,t_COL);
    1441        3402 :   for (i=1; i<=n; i++)
    1442             :   {
    1443        2989 :     GEN mi = gel(mt,i);
    1444       34790 :     for (j=1; j<=n; j++) gcoeff(M,i,j) = gel(mi,j); /* ei.ej */
    1445             :   }
    1446        2975 :   for (i=2; i<=n; i++) {
    1447        2569 :     GEN mi = gel(mt,i);
    1448       28777 :     for (j=2; j<=n; j++) {
    1449      367759 :       for (k=2; k<=n; k++) {
    1450             :         GEN x, y;
    1451      341551 :         if (signe(p)) {
    1452      242039 :           x = _tablemul_ej_Fp(mt,gcoeff(M,i,j),k,p);
    1453      242039 :           y = FpM_FpC_mul(mi,gcoeff(M,j,k),p);
    1454             :         }
    1455             :         else {
    1456       99512 :           x = _tablemul_ej(mt,gcoeff(M,i,j),k);
    1457       99512 :           y = RgM_RgC_mul(mi,gcoeff(M,j,k));
    1458             :         }
    1459             :         /* not cmp_universal: must not fail on 0 == Mod(0,2) for instance */
    1460      341551 :         if (!gequal(x,y)) return gc_bool(av,0);
    1461             :       }
    1462             :     }
    1463             :   }
    1464         406 :   return gc_bool(av,1);
    1465             : }
    1466             : 
    1467             : int
    1468         350 : algiscommutative(GEN al) /* assumes e_1 = 1 */
    1469             : {
    1470             :   long i,j,k,N,sp;
    1471             :   GEN mt,a,b,p;
    1472         350 :   checkalg(al);
    1473         350 :   if (alg_type(al) != al_TABLE) return alg_get_degree(al)==1;
    1474         308 :   N = alg_get_absdim(al);
    1475         308 :   mt = alg_get_multable(al);
    1476         308 :   p = alg_get_char(al);
    1477         308 :   sp = signe(p);
    1478        1449 :   for (i=2; i<=N; i++)
    1479        9464 :     for (j=2; j<=N; j++)
    1480       85820 :       for (k=1; k<=N; k++) {
    1481       77553 :         a = gcoeff(gel(mt,i),k,j);
    1482       77553 :         b = gcoeff(gel(mt,j),k,i);
    1483       77553 :         if (sp) {
    1484       73423 :           if (cmpii(Fp_red(a,p), Fp_red(b,p))) return 0;
    1485             :         }
    1486        4130 :         else if (gcmp(a,b)) return 0;
    1487             :       }
    1488         252 :   return 1;
    1489             : }
    1490             : 
    1491             : int
    1492         350 : algissemisimple(GEN al)
    1493             : {
    1494         350 :   pari_sp av = avma;
    1495             :   GEN rad;
    1496         350 :   checkalg(al);
    1497         350 :   if (alg_type(al) != al_TABLE) return 1;
    1498         308 :   rad = algradical(al);
    1499         308 :   set_avma(av);
    1500         308 :   return gequal0(rad);
    1501             : }
    1502             : 
    1503             : /* ss : known to be semisimple */
    1504             : int
    1505         259 : algissimple(GEN al, long ss)
    1506             : {
    1507         259 :   pari_sp av = avma;
    1508             :   GEN Z, dec, p;
    1509         259 :   checkalg(al);
    1510         259 :   if (alg_type(al) != al_TABLE) return 1;
    1511         224 :   if (!ss && !algissemisimple(al)) return 0;
    1512             : 
    1513         182 :   p = alg_get_char(al);
    1514         182 :   if (signe(p)) Z = algprimesubalg(al);
    1515          91 :   else          Z = algtablecenter(al);
    1516             : 
    1517         182 :   if (lg(Z) == 2) {/* dim Z = 1 */
    1518         105 :     set_avma(av);
    1519         105 :     return 1;
    1520             :   }
    1521          77 :   dec = alg_decompose(al, Z, 1, NULL);
    1522          77 :   set_avma(av);
    1523          77 :   return gequal0(dec);
    1524             : }
    1525             : 
    1526             : static long
    1527         329 : is_place_emb(GEN nf, GEN pl)
    1528             : {
    1529             :   long r, r1, r2;
    1530         329 :   if (typ(pl) != t_INT) pari_err_TYPE("is_place_emb", pl);
    1531         315 :   if (signe(pl)<=0) pari_err_DOMAIN("is_place_emb", "pl", "<=", gen_0, pl);
    1532         308 :   nf_get_sign(nf,&r1,&r2); r = r1+r2;
    1533         308 :   if (cmpiu(pl,r)>0) pari_err_DOMAIN("is_place_emb", "pl", ">", utoi(r), pl);
    1534         294 :   return itou(pl);
    1535             : }
    1536             : 
    1537             : static long
    1538         294 : alghasse_emb(GEN al, long emb)
    1539             : {
    1540         294 :   GEN nf = alg_get_center(al);
    1541         294 :   long r1 = nf_get_r1(nf);
    1542         294 :   return (emb <= r1)? alg_get_hasse_i(al)[emb]: 0;
    1543             : }
    1544             : 
    1545             : static long
    1546         399 : alghasse_pr(GEN al, GEN pr)
    1547             : {
    1548         399 :   GEN hf = alg_get_hasse_f(al);
    1549         399 :   long i = tablesearch(gel(hf,1), pr, &cmp_prime_ideal);
    1550         399 :   return i? gel(hf,2)[i]: 0;
    1551             : }
    1552             : 
    1553             : static long
    1554         735 : alghasse_0(GEN al, GEN pl)
    1555             : {
    1556             :   GEN pr, nf;
    1557         735 :   if (alg_type(al)== al_CSA)
    1558           7 :     pari_err_IMPL("computation of Hasse invariants over table CSA");
    1559         728 :   if ((pr = get_prid(pl))) return alghasse_pr(al, pr);
    1560         329 :   nf = alg_get_center(al);
    1561         329 :   return alghasse_emb(al, is_place_emb(nf, pl));
    1562             : }
    1563             : GEN
    1564         210 : alghasse(GEN al, GEN pl)
    1565             : {
    1566             :   long h;
    1567         210 :   checkalg(al);
    1568         210 :   if (alg_type(al) == al_TABLE) pari_err_TYPE("alghasse [use alginit]",al);
    1569         203 :   h = alghasse_0(al,pl);
    1570         161 :   return sstoQ(h, alg_get_degree(al));
    1571             : }
    1572             : 
    1573             : /* h >= 0, d >= 0 */
    1574             : static long
    1575         812 : indexfromhasse(long h, long d) { return d/ugcd(h,d); }
    1576             : 
    1577             : long
    1578         728 : algindex(GEN al, GEN pl)
    1579             : {
    1580             :   long d, res, i, l;
    1581             :   GEN hi, hf;
    1582             : 
    1583         728 :   checkalg(al);
    1584         721 :   if (alg_type(al) == al_TABLE) pari_err_TYPE("algindex [use alginit]",al);
    1585         714 :   d = alg_get_degree(al);
    1586         714 :   if (pl) return indexfromhasse(alghasse_0(al,pl), d);
    1587             : 
    1588             :   /* else : global index */
    1589         182 :   res = 1;
    1590         182 :   hi = alg_get_hasse_i(al); l = lg(hi);
    1591         308 :   for (i=1; i<l && res!=d; i++) res = ulcm(res, indexfromhasse(hi[i],d));
    1592         182 :   hf = gel(alg_get_hasse_f(al), 2); l = lg(hf);
    1593         336 :   for (i=1; i<l && res!=d; i++) res = ulcm(res, indexfromhasse(hf[i],d));
    1594         182 :   return res;
    1595             : }
    1596             : 
    1597             : int
    1598         203 : algisdivision(GEN al, GEN pl)
    1599             : {
    1600         203 :   checkalg(al);
    1601         203 :   if (alg_type(al) == al_TABLE) {
    1602          21 :     if (!algissimple(al,0)) return 0;
    1603          14 :     if (algiscommutative(al)) return 1;
    1604           7 :     pari_err_IMPL("algisdivision for table algebras");
    1605             :   }
    1606         182 :   return algindex(al,pl) == alg_get_degree(al);
    1607             : }
    1608             : 
    1609             : int
    1610         182 : algissplit(GEN al, GEN pl)
    1611             : {
    1612         182 :   checkalg(al);
    1613         182 :   if (alg_type(al) == al_TABLE) pari_err_TYPE("algissplit [use alginit]", al);
    1614         175 :   return algindex(al,pl) == 1;
    1615             : }
    1616             : 
    1617             : int
    1618         182 : algisramified(GEN al, GEN pl)
    1619             : {
    1620         182 :   checkalg(al);
    1621         182 :   if (alg_type(al) == al_TABLE) pari_err_TYPE("algisramified [use alginit]", al);
    1622         175 :   return algindex(al,pl) != 1;
    1623             : }
    1624             : 
    1625             : GEN
    1626          91 : algramifiedplaces(GEN al)
    1627             : {
    1628          91 :   pari_sp av = avma;
    1629             :   GEN ram, hf, hi, Lpr;
    1630             :   long r1, count, i;
    1631          91 :   checkalg(al);
    1632          91 :   if (alg_type(al) == al_TABLE) pari_err_TYPE("algramifiedplaces [use alginit]", al);
    1633          84 :   r1 = nf_get_r1(alg_get_center(al));
    1634          84 :   hi = alg_get_hasse_i(al);
    1635          84 :   hf = alg_get_hasse_f(al);
    1636          84 :   Lpr = gel(hf,1);
    1637          84 :   hf = gel(hf,2);
    1638          84 :   ram = cgetg(r1+lg(Lpr), t_VEC);
    1639          84 :   count = 0;
    1640         280 :   for (i=1; i<=r1; i++)
    1641         196 :     if (hi[i]) {
    1642          91 :       count++;
    1643          91 :       gel(ram,count) = stoi(i);
    1644             :     }
    1645         280 :   for (i=1; i<lg(Lpr); i++)
    1646         196 :     if (hf[i]) {
    1647          77 :       count++;
    1648          77 :       gel(ram,count) = gel(Lpr,i);
    1649             :     }
    1650          84 :   setlg(ram, count+1);
    1651          84 :   return gerepilecopy(av, ram);
    1652             : }
    1653             : 
    1654             : /** OPERATIONS ON ELEMENTS operations.c **/
    1655             : 
    1656             : static long
    1657     1043107 : alg_model0(GEN al, GEN x)
    1658             : {
    1659     1043107 :   long t, N = alg_get_absdim(al), lx = lg(x), d, n, D, i;
    1660     1043107 :   if (typ(x) == t_MAT) return al_MATRIX;
    1661      997152 :   if (typ(x) != t_COL) return al_INVALID;
    1662      997089 :   if (N == 1) {
    1663        2667 :     if (lx != 2) return al_INVALID;
    1664        2646 :     switch(typ(gel(x,1)))
    1665             :     {
    1666        1652 :       case t_INT: case t_FRAC: return al_TRIVIAL; /* cannot distinguish basis and alg from size */
    1667         994 :       case t_POL: case t_POLMOD: return al_ALGEBRAIC;
    1668           0 :       default: return al_INVALID;
    1669             :     }
    1670             :   }
    1671             : 
    1672      994422 :   switch(alg_type(al)) {
    1673      549824 :     case al_TABLE:
    1674      549824 :       if (lx != N+1) return al_INVALID;
    1675      549803 :       return al_BASIS;
    1676      358554 :     case al_CYCLIC:
    1677      358554 :       d = alg_get_degree(al);
    1678      358554 :       if (lx == N+1) return al_BASIS;
    1679      100667 :       if (lx == d+1) return al_ALGEBRAIC;
    1680          14 :       return al_INVALID;
    1681       86044 :     case al_CSA:
    1682       86044 :       D = alg_get_dim(al);
    1683       86044 :       n = nf_get_degree(alg_get_center(al));
    1684       86044 :       if (n == 1) {
    1685        1302 :         if (lx != D+1) return al_INVALID;
    1686        3871 :         for (i=1; i<=D; i++) {
    1687        3227 :           t = typ(gel(x,i));
    1688        3227 :           if (t == t_POL || t == t_POLMOD)  return al_ALGEBRAIC;
    1689             :             /* TODO t_COL for coefficients in basis form ? */
    1690             :         }
    1691         644 :         return al_BASIS;
    1692             :       }
    1693             :       else {
    1694       84742 :         if (lx == N+1) return al_BASIS;
    1695       23135 :         if (lx == D+1) return al_ALGEBRAIC;
    1696           0 :         return al_INVALID;
    1697             :       }
    1698             :   }
    1699             :   return al_INVALID; /* LCOV_EXCL_LINE */
    1700             : }
    1701             : 
    1702             : static void
    1703     1042981 : checkalgx(GEN x, long model)
    1704             : {
    1705             :   long t, i;
    1706     1042981 :   switch(model) {
    1707      869941 :     case al_BASIS:
    1708     9176424 :       for (i=1; i<lg(x); i++) {
    1709     8306490 :         t = typ(gel(x,i));
    1710     8306490 :         if (t != t_INT && t != t_FRAC)
    1711           7 :           pari_err_TYPE("checkalgx", gel(x,i));
    1712             :       }
    1713      869934 :       return;
    1714      127085 :     case al_TRIVIAL:
    1715             :     case al_ALGEBRAIC:
    1716      444171 :       for (i=1; i<lg(x); i++) {
    1717      317093 :         t = typ(gel(x,i));
    1718      317093 :         if (t != t_INT && t != t_FRAC && t != t_POL && t != t_POLMOD)
    1719             :           /* TODO t_COL ? */
    1720           7 :           pari_err_TYPE("checkalgx", gel(x,i));
    1721             :       }
    1722      127078 :       return;
    1723             :   }
    1724             : }
    1725             : 
    1726             : long
    1727     1043107 : alg_model(GEN al, GEN x)
    1728             : {
    1729     1043107 :   long res = alg_model0(al, x);
    1730     1043107 :   if (res == al_INVALID) pari_err_TYPE("alg_model", x);
    1731     1042981 :   checkalgx(x, res); return res;
    1732             : }
    1733             : 
    1734             : static GEN
    1735         518 : alC_add_i(GEN al, GEN x, GEN y, long lx)
    1736             : {
    1737         518 :   GEN A = cgetg(lx, t_COL);
    1738             :   long i;
    1739        1554 :   for (i=1; i<lx; i++) gel(A,i) = algadd(al, gel(x,i), gel(y,i));
    1740         518 :   return A;
    1741             : }
    1742             : static GEN
    1743         280 : alM_add(GEN al, GEN x, GEN y)
    1744             : {
    1745         280 :   long lx = lg(x), l, j;
    1746             :   GEN z;
    1747         280 :   if (lg(y) != lx) pari_err_DIM("alM_add (rows)");
    1748         273 :   if (lx == 1) return cgetg(1, t_MAT);
    1749         266 :   z = cgetg(lx, t_MAT); l = lgcols(x);
    1750         266 :   if (lgcols(y) != l) pari_err_DIM("alM_add (columns)");
    1751         777 :   for (j = 1; j < lx; j++) gel(z,j) = alC_add_i(al, gel(x,j), gel(y,j), l);
    1752         259 :   return z;
    1753             : }
    1754             : GEN
    1755       36974 : algadd(GEN al, GEN x, GEN y)
    1756             : {
    1757       36974 :   pari_sp av = avma;
    1758             :   long tx, ty;
    1759             :   GEN p;
    1760       36974 :   checkalg(al);
    1761       36974 :   tx = alg_model(al,x);
    1762       36967 :   ty = alg_model(al,y);
    1763       36967 :   p = alg_get_char(al);
    1764       36967 :   if (signe(p)) return FpC_add(x,y,p);
    1765       36834 :   if (tx==ty) {
    1766       36022 :     if (tx!=al_MATRIX) return gadd(x,y);
    1767         280 :     return gerepilecopy(av, alM_add(al,x,y));
    1768             :   }
    1769         812 :   if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
    1770         812 :   if (ty==al_ALGEBRAIC) y = algalgtobasis(al,y);
    1771         812 :   return gerepileupto(av, gadd(x,y));
    1772             : }
    1773             : 
    1774             : GEN
    1775         147 : algneg(GEN al, GEN x) { checkalg(al); (void)alg_model(al,x); return gneg(x); }
    1776             : 
    1777             : static GEN
    1778         210 : alC_sub_i(GEN al, GEN x, GEN y, long lx)
    1779             : {
    1780             :   long i;
    1781         210 :   GEN A = cgetg(lx, t_COL);
    1782         630 :   for (i=1; i<lx; i++) gel(A,i) = algsub(al, gel(x,i), gel(y,i));
    1783         210 :   return A;
    1784             : }
    1785             : static GEN
    1786         126 : alM_sub(GEN al, GEN x, GEN y)
    1787             : {
    1788         126 :   long lx = lg(x), l, j;
    1789             :   GEN z;
    1790         126 :   if (lg(y) != lx) pari_err_DIM("alM_sub (rows)");
    1791         119 :   if (lx == 1) return cgetg(1, t_MAT);
    1792         112 :   z = cgetg(lx, t_MAT); l = lgcols(x);
    1793         112 :   if (lgcols(y) != l) pari_err_DIM("alM_sub (columns)");
    1794         315 :   for (j = 1; j < lx; j++) gel(z,j) = alC_sub_i(al, gel(x,j), gel(y,j), l);
    1795         105 :   return z;
    1796             : }
    1797             : GEN
    1798         966 : algsub(GEN al, GEN x, GEN y)
    1799             : {
    1800             :   long tx, ty;
    1801         966 :   pari_sp av = avma;
    1802             :   GEN p;
    1803         966 :   checkalg(al);
    1804         966 :   tx = alg_model(al,x);
    1805         959 :   ty = alg_model(al,y);
    1806         959 :   p = alg_get_char(al);
    1807         959 :   if (signe(p)) return FpC_sub(x,y,p);
    1808         868 :   if (tx==ty) {
    1809         546 :     if (tx != al_MATRIX) return gsub(x,y);
    1810         126 :     return gerepilecopy(av, alM_sub(al,x,y));
    1811             :   }
    1812         322 :   if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
    1813         322 :   if (ty==al_ALGEBRAIC) y = algalgtobasis(al,y);
    1814         322 :   return gerepileupto(av, gsub(x,y));
    1815             : }
    1816             : 
    1817             : static GEN
    1818        1659 : algalgmul_cyc(GEN al, GEN x, GEN y)
    1819             : {
    1820        1659 :   pari_sp av = avma;
    1821        1659 :   long n = alg_get_degree(al), i, k;
    1822             :   GEN xalg, yalg, res, rnf, auts, sum, b, prod, autx;
    1823        1659 :   rnf = alg_get_splittingfield(al);
    1824        1659 :   auts = alg_get_auts(al);
    1825        1659 :   b = alg_get_b(al);
    1826             : 
    1827        1659 :   xalg = cgetg(n+1, t_COL);
    1828        4935 :   for (i=0; i<n; i++)
    1829        3276 :     gel(xalg,i+1) = lift_shallow(rnfbasistoalg(rnf,gel(x,i+1)));
    1830             : 
    1831        1659 :   yalg = cgetg(n+1, t_COL);
    1832        4935 :   for (i=0; i<n; i++) gel(yalg,i+1) = rnfbasistoalg(rnf,gel(y,i+1));
    1833             : 
    1834        1659 :   res = cgetg(n+1,t_COL);
    1835        4935 :   for (k=0; k<n; k++) {
    1836        3276 :     gel(res,k+1) = gmul(gel(xalg,k+1),gel(yalg,1));
    1837        5166 :     for (i=1; i<=k; i++) {
    1838        1890 :       autx = poleval(gel(xalg,k-i+1),gel(auts,i));
    1839        1890 :       prod = gmul(autx,gel(yalg,i+1));
    1840        1890 :       gel(res,k+1) = gadd(gel(res,k+1), prod);
    1841             :     }
    1842             : 
    1843        3276 :     sum = gen_0;
    1844        5166 :     for (; i<n; i++) {
    1845        1890 :       autx = poleval(gel(xalg,k+n-i+1),gel(auts,i));
    1846        1890 :       prod = gmul(autx,gel(yalg,i+1));
    1847        1890 :       sum = gadd(sum,prod);
    1848             :     }
    1849        3276 :     sum = gmul(b,sum);
    1850             : 
    1851        3276 :     gel(res,k+1) = gadd(gel(res,k+1),sum);
    1852             :   }
    1853             : 
    1854        1659 :   return gerepilecopy(av, res);
    1855             : }
    1856             : 
    1857             : static GEN
    1858      203763 : _tablemul(GEN mt, GEN x, GEN y)
    1859             : {
    1860      203763 :   pari_sp av = avma;
    1861      203763 :   long D = lg(mt)-1, i;
    1862      203763 :   GEN res = NULL;
    1863     1906905 :   for (i=1; i<=D; i++) {
    1864     1703142 :     GEN c = gel(x,i);
    1865     1703142 :     if (!gequal0(c)) {
    1866      988876 :       GEN My = RgM_RgC_mul(gel(mt,i),y);
    1867      988876 :       GEN t = RgC_Rg_mul(My,c);
    1868      988876 :       res = res? RgC_add(res,t): t;
    1869             :     }
    1870             :   }
    1871      203763 :   if (!res) { set_avma(av); return zerocol(D); }
    1872      202860 :   return gerepileupto(av, res);
    1873             : }
    1874             : 
    1875             : static GEN
    1876      189843 : _tablemul_Fp(GEN mt, GEN x, GEN y, GEN p)
    1877             : {
    1878      189843 :   pari_sp av = avma;
    1879      189843 :   long D = lg(mt)-1, i;
    1880      189843 :   GEN res = NULL;
    1881     2238261 :   for (i=1; i<=D; i++) {
    1882     2048418 :     GEN c = gel(x,i);
    1883     2048418 :     if (signe(c)) {
    1884      325839 :       GEN My = FpM_FpC_mul(gel(mt,i),y,p);
    1885      325839 :       GEN t = FpC_Fp_mul(My,c,p);
    1886      325839 :       res = res? FpC_add(res,t,p): t;
    1887             :     }
    1888             :   }
    1889      189843 :   if (!res) { set_avma(av); return zerocol(D); }
    1890      189304 :   return gerepileupto(av, res);
    1891             : }
    1892             : 
    1893             : /* x*ej */
    1894             : static GEN
    1895       99512 : _tablemul_ej(GEN mt, GEN x, long j)
    1896             : {
    1897       99512 :   pari_sp av = avma;
    1898       99512 :   long D = lg(mt)-1, i;
    1899       99512 :   GEN res = NULL;
    1900     1561861 :   for (i=1; i<=D; i++) {
    1901     1462349 :     GEN c = gel(x,i);
    1902     1462349 :     if (!gequal0(c)) {
    1903      114023 :       GEN My = gel(gel(mt,i),j);
    1904      114023 :       GEN t = RgC_Rg_mul(My,c);
    1905      114023 :       res = res? RgC_add(res,t): t;
    1906             :     }
    1907             :   }
    1908       99512 :   if (!res) { set_avma(av); return zerocol(D); }
    1909       99372 :   return gerepileupto(av, res);
    1910             : }
    1911             : static GEN
    1912      242039 : _tablemul_ej_Fp(GEN mt, GEN x, long j, GEN p)
    1913             : {
    1914      242039 :   pari_sp av = avma;
    1915      242039 :   long D = lg(mt)-1, i;
    1916      242039 :   GEN res = NULL;
    1917     4364787 :   for (i=1; i<=D; i++) {
    1918     4122748 :     GEN c = gel(x,i);
    1919     4122748 :     if (!gequal0(c)) {
    1920      289954 :       GEN My = gel(gel(mt,i),j);
    1921      289954 :       GEN t = FpC_Fp_mul(My,c,p);
    1922      289954 :       res = res? FpC_add(res,t,p): t;
    1923             :     }
    1924             :   }
    1925      242039 :   if (!res) { set_avma(av); return zerocol(D); }
    1926      241927 :   return gerepileupto(av, res);
    1927             : }
    1928             : 
    1929             : static GEN
    1930      243523 : _tablemul_ej_Fl(GEN mt, GEN x, long j, ulong p)
    1931             : {
    1932      243523 :   pari_sp av = avma;
    1933      243523 :   long D = lg(mt)-1, i;
    1934      243523 :   GEN res = NULL;
    1935     3937416 :   for (i=1; i<=D; i++) {
    1936     3693893 :     ulong c = x[i];
    1937     3693893 :     if (c) {
    1938      385350 :       GEN My = gel(gel(mt,i),j);
    1939      385350 :       GEN t = Flv_Fl_mul(My,c, p);
    1940      385350 :       res = res? Flv_add(res,t, p): t;
    1941             :     }
    1942             :   }
    1943      243523 :   if (!res) { set_avma(av); return zero_Flv(D); }
    1944      243523 :   return gerepileupto(av, res);
    1945             : }
    1946             : 
    1947             : static GEN
    1948         686 : algalgmul_csa(GEN al, GEN x, GEN y)
    1949             : {
    1950         686 :   GEN z, nf = alg_get_center(al);
    1951             :   long i;
    1952         686 :   z = _tablemul(alg_get_relmultable(al), x, y);
    1953        2485 :   for (i=1; i<lg(z); i++)
    1954        1799 :     gel(z,i) = basistoalg(nf,gel(z,i));
    1955         686 :   return z;
    1956             : }
    1957             : 
    1958             : /* assumes x and y in algebraic form */
    1959             : static GEN
    1960        2345 : algalgmul(GEN al, GEN x, GEN y)
    1961             : {
    1962        2345 :   switch(alg_type(al))
    1963             :   {
    1964        1659 :     case al_CYCLIC: return algalgmul_cyc(al, x, y);
    1965         686 :     case al_CSA: return algalgmul_csa(al, x, y);
    1966             :   }
    1967             :   return NULL; /*LCOV_EXCL_LINE*/
    1968             : }
    1969             : 
    1970             : static GEN
    1971      392920 : algbasismul(GEN al, GEN x, GEN y)
    1972             : {
    1973      392920 :   GEN mt = alg_get_multable(al), p = alg_get_char(al);
    1974      392920 :   if (signe(p)) return _tablemul_Fp(mt, x, y, p);
    1975      203077 :   return _tablemul(mt, x, y);
    1976             : }
    1977             : 
    1978             : /* x[i,]*y. Assume lg(x) > 1 and 0 < i < lgcols(x) */
    1979             : static GEN
    1980       85001 : alMrow_alC_mul_i(GEN al, GEN x, GEN y, long i, long lx)
    1981             : {
    1982       85001 :   pari_sp av = avma;
    1983       85001 :   GEN c = algmul(al,gcoeff(x,i,1),gel(y,1)), ZERO;
    1984             :   long k;
    1985       85001 :   ZERO = zerocol(alg_get_absdim(al));
    1986      170002 :   for (k = 2; k < lx; k++)
    1987             :   {
    1988       85001 :     GEN t = algmul(al, gcoeff(x,i,k), gel(y,k));
    1989       85001 :     if (!gequal(t,ZERO)) c = algadd(al, c, t);
    1990             :   }
    1991       85001 :   return gerepilecopy(av, c);
    1992             : }
    1993             : /* return x * y, 1 < lx = lg(x), l = lgcols(x) */
    1994             : static GEN
    1995       42518 : alM_alC_mul_i(GEN al, GEN x, GEN y, long lx, long l)
    1996             : {
    1997       42518 :   GEN z = cgetg(l,t_COL);
    1998             :   long i;
    1999      127519 :   for (i=1; i<l; i++) gel(z,i) = alMrow_alC_mul_i(al,x,y,i,lx);
    2000       42518 :   return z;
    2001             : }
    2002             : static GEN
    2003       21336 : alM_mul(GEN al, GEN x, GEN y)
    2004             : {
    2005       21336 :   long j, l, lx=lg(x), ly=lg(y);
    2006             :   GEN z;
    2007       21336 :   if (ly==1) return cgetg(1,t_MAT);
    2008       21287 :   if (lx != lgcols(y)) pari_err_DIM("alM_mul");
    2009       21266 :   if (lx==1) return zeromat(0, ly-1);
    2010       21259 :   l = lgcols(x); z = cgetg(ly,t_MAT);
    2011       63777 :   for (j=1; j<ly; j++) gel(z,j) = alM_alC_mul_i(al,x,gel(y,j),lx,l);
    2012       21259 :   return z;
    2013             : }
    2014             : 
    2015             : GEN
    2016      365165 : algmul(GEN al, GEN x, GEN y)
    2017             : {
    2018      365165 :   pari_sp av = avma;
    2019             :   long tx, ty;
    2020      365165 :   checkalg(al);
    2021      365165 :   tx = alg_model(al,x);
    2022      365151 :   ty = alg_model(al,y);
    2023      365151 :   if (tx==al_MATRIX) {
    2024       20832 :     if (ty==al_MATRIX) return alM_mul(al,x,y);
    2025           7 :     pari_err_TYPE("algmul", y);
    2026             :   }
    2027      344319 :   if (signe(alg_get_char(al))) return algbasismul(al,x,y);
    2028      203504 :   if (tx==al_TRIVIAL) retmkcol(gmul(gel(x,1),gel(y,1)));
    2029      203399 :   if (tx==al_ALGEBRAIC && ty==al_ALGEBRAIC) return algalgmul(al,x,y);
    2030      201873 :   if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
    2031      201873 :   if (ty==al_ALGEBRAIC) y = algalgtobasis(al,y);
    2032      201873 :   return gerepileupto(av,algbasismul(al,x,y));
    2033             : }
    2034             : 
    2035             : GEN
    2036       48804 : algsqr(GEN al, GEN x)
    2037             : {
    2038       48804 :   pari_sp av = avma;
    2039             :   long tx;
    2040       48804 :   checkalg(al);
    2041       48769 :   tx = alg_model(al,x);
    2042       48713 :   if (tx==al_MATRIX) return gerepilecopy(av,alM_mul(al,x,x));
    2043       48202 :   if (signe(alg_get_char(al))) return algbasismul(al,x,x);
    2044        2205 :   if (tx==al_TRIVIAL) retmkcol(gsqr(gel(x,1)));
    2045        2023 :   if (tx==al_ALGEBRAIC) return algalgmul(al,x,x);
    2046        1204 :   return gerepileupto(av,algbasismul(al,x,x));
    2047             : }
    2048             : 
    2049             : static GEN
    2050        8099 : algmtK2Z_cyc(GEN al, GEN m)
    2051             : {
    2052        8099 :   pari_sp av = avma;
    2053        8099 :   GEN nf = alg_get_abssplitting(al), res, mt, rnf = alg_get_splittingfield(al), c, dc;
    2054        8099 :   long n = alg_get_degree(al), N = nf_get_degree(nf), Nn, i, j, i1, j1;
    2055        8099 :   Nn = N*n;
    2056        8099 :   res = zeromatcopy(Nn,Nn);
    2057       38150 :   for (i=0; i<n; i++)
    2058      186242 :   for (j=0; j<n; j++) {
    2059      156191 :     c = gcoeff(m,i+1,j+1);
    2060      156191 :     if (!gequal0(c)) {
    2061       30051 :       c = rnfeltreltoabs(rnf,c);
    2062       30051 :       c = algtobasis(nf,c);
    2063       30051 :       c = Q_remove_denom(c,&dc);
    2064       30051 :       mt = zk_multable(nf,c);
    2065       30051 :       if (dc) mt = ZM_Z_div(mt,dc);
    2066      270634 :       for (i1=1; i1<=N; i1++)
    2067     2529646 :       for (j1=1; j1<=N; j1++)
    2068     2289063 :         gcoeff(res,i*N+i1,j*N+j1) = gcoeff(mt,i1,j1);
    2069             :     }
    2070             :   }
    2071        8099 :   return gerepilecopy(av,res);
    2072             : }
    2073             : 
    2074             : static GEN
    2075         861 : algmtK2Z_csa(GEN al, GEN m)
    2076             : {
    2077         861 :   pari_sp av = avma;
    2078         861 :   GEN nf = alg_get_center(al), res, mt, c, dc;
    2079         861 :   long d2 = alg_get_dim(al), n = nf_get_degree(nf), D, i, j, i1, j1;
    2080         861 :   D = d2*n;
    2081         861 :   res = zeromatcopy(D,D);
    2082        5082 :   for (i=0; i<d2; i++)
    2083       29442 :   for (j=0; j<d2; j++) {
    2084       25221 :     c = gcoeff(m,i+1,j+1);
    2085       25221 :     if (!gequal0(c)) {
    2086        3360 :       c = algtobasis(nf,c);
    2087        3360 :       c = Q_remove_denom(c,&dc);
    2088        3360 :       mt = zk_multable(nf,c);
    2089        3360 :       if (dc) mt = ZM_Z_div(mt,dc);
    2090       11550 :       for (i1=1; i1<=n; i1++)
    2091       29736 :       for (j1=1; j1<=n; j1++)
    2092       21546 :         gcoeff(res,i*n+i1,j*n+j1) = gcoeff(mt,i1,j1);
    2093             :     }
    2094             :   }
    2095         861 :   return gerepilecopy(av,res);
    2096             : }
    2097             : 
    2098             : /* assumes al is a CSA or CYCLIC */
    2099             : static GEN
    2100        8960 : algmtK2Z(GEN al, GEN m)
    2101             : {
    2102        8960 :   switch(alg_type(al))
    2103             :   {
    2104        8099 :     case al_CYCLIC: return algmtK2Z_cyc(al, m);
    2105         861 :     case al_CSA: return algmtK2Z_csa(al, m);
    2106             :   }
    2107             :   return NULL; /*LCOV_EXCL_LINE*/
    2108             : }
    2109             : 
    2110             : /* left multiplication table, as a vector space of dimension n over the splitting field (by right multiplication) */
    2111             : static GEN
    2112       10717 : algalgmultable_cyc(GEN al, GEN x)
    2113             : {
    2114       10717 :   pari_sp av = avma;
    2115       10717 :   long n = alg_get_degree(al), i, j;
    2116             :   GEN res, rnf, auts, b, pol;
    2117       10717 :   rnf = alg_get_splittingfield(al);
    2118       10717 :   auts = alg_get_auts(al);
    2119       10717 :   b = alg_get_b(al);
    2120       10717 :   pol = rnf_get_pol(rnf);
    2121             : 
    2122       10717 :   res = zeromatcopy(n,n);
    2123       46074 :   for (i=0; i<n; i++)
    2124       35357 :     gcoeff(res,i+1,1) = lift_shallow(rnfbasistoalg(rnf,gel(x,i+1)));
    2125             : 
    2126       46074 :   for (i=0; i<n; i++) {
    2127      101423 :     for (j=1; j<=i; j++)
    2128       66066 :       gcoeff(res,i+1,j+1) = gmodulo(poleval(gcoeff(res,i-j+1,1),gel(auts,j)),pol);
    2129      101423 :     for (; j<n; j++)
    2130       66066 :       gcoeff(res,i+1,j+1) = gmodulo(gmul(b,poleval(gcoeff(res,n+i-j+1,1),gel(auts,j))), pol);
    2131             :   }
    2132             : 
    2133       46074 :   for (i=0; i<n; i++)
    2134       35357 :     gcoeff(res,i+1,1) = gmodulo(gcoeff(res,i+1,1),pol);
    2135             : 
    2136       10717 :   return gerepilecopy(av, res);
    2137             : }
    2138             : 
    2139             : static GEN
    2140        1309 : elementmultable(GEN mt, GEN x)
    2141             : {
    2142        1309 :   pari_sp av = avma;
    2143        1309 :   long D = lg(mt)-1, i;
    2144        1309 :   GEN z = NULL;
    2145        7028 :   for (i=1; i<=D; i++)
    2146             :   {
    2147        5719 :     GEN c = gel(x,i);
    2148        5719 :     if (!gequal0(c))
    2149             :     {
    2150        2079 :       GEN M = RgM_Rg_mul(gel(mt,i),c);
    2151        2079 :       z = z? RgM_add(z, M): M;
    2152             :     }
    2153             :   }
    2154        1309 :   if (!z) { set_avma(av); return zeromatcopy(D,D); }
    2155        1309 :   return gerepileupto(av, z);
    2156             : }
    2157             : /* mt a t_VEC of Flm modulo m */
    2158             : static GEN
    2159       24472 : algbasismultable_Flm(GEN mt, GEN x, ulong m)
    2160             : {
    2161       24472 :   pari_sp av = avma;
    2162       24472 :   long D = lg(gel(mt,1))-1, i;
    2163       24472 :   GEN z = NULL;
    2164      267995 :   for (i=1; i<=D; i++)
    2165             :   {
    2166      243523 :     ulong c = x[i];
    2167      243523 :     if (c)
    2168             :     {
    2169       33390 :       GEN M = Flm_Fl_mul(gel(mt,i),c, m);
    2170       33390 :       z = z? Flm_add(z, M, m): M;
    2171             :     }
    2172             :   }
    2173       24472 :   if (!z) { set_avma(av); return zero_Flm(D,D); }
    2174       24472 :   return gerepileupto(av, z);
    2175             : }
    2176             : static GEN
    2177      220413 : elementabsmultable_Z(GEN mt, GEN x)
    2178             : {
    2179      220413 :   long i, l = lg(x);
    2180      220413 :   GEN z = NULL;
    2181     2378925 :   for (i = 1; i < l; i++)
    2182             :   {
    2183     2158512 :     GEN c = gel(x,i);
    2184     2158512 :     if (signe(c))
    2185             :     {
    2186      839011 :       GEN M = ZM_Z_mul(gel(mt,i),c);
    2187      839011 :       z = z? ZM_add(z, M): M;
    2188             :     }
    2189             :   }
    2190      220413 :   return z;
    2191             : }
    2192             : static GEN
    2193      114660 : elementabsmultable(GEN mt, GEN x)
    2194             : {
    2195      114660 :   GEN d, z = elementabsmultable_Z(mt, Q_remove_denom(x,&d));
    2196      114660 :   return (z && d)? ZM_Z_div(z, d): z;
    2197             : }
    2198             : static GEN
    2199      105753 : elementabsmultable_Fp(GEN mt, GEN x, GEN p)
    2200             : {
    2201      105753 :   GEN z = elementabsmultable_Z(mt, x);
    2202      105753 :   return z? FpM_red(z, p): z;
    2203             : }
    2204             : static GEN
    2205      220413 : algbasismultable(GEN al, GEN x)
    2206             : {
    2207      220413 :   pari_sp av = avma;
    2208      220413 :   GEN z, p = alg_get_char(al), mt = alg_get_multable(al);
    2209      220413 :   z = signe(p)? elementabsmultable_Fp(mt, x, p): elementabsmultable(mt, x);
    2210      220413 :   if (!z)
    2211             :   {
    2212         761 :     long D = lg(mt)-1;
    2213         761 :     set_avma(av); return zeromat(D,D);
    2214             :   }
    2215      219652 :   return gerepileupto(av, z);
    2216             : }
    2217             : 
    2218             : static GEN
    2219        1309 : algalgmultable_csa(GEN al, GEN x)
    2220             : {
    2221        1309 :   GEN nf = alg_get_center(al), m;
    2222             :   long i,j;
    2223        1309 :   m = elementmultable(alg_get_relmultable(al), x);
    2224        7028 :   for (i=1; i<lg(m); i++)
    2225       36638 :     for(j=1; j<lg(m); j++)
    2226       30919 :       gcoeff(m,i,j) = basistoalg(nf,gcoeff(m,i,j));
    2227        1309 :   return m;
    2228             : }
    2229             : 
    2230             : /* assumes x in algebraic form */
    2231             : static GEN
    2232       11732 : algalgmultable(GEN al, GEN x)
    2233             : {
    2234       11732 :   switch(alg_type(al))
    2235             :   {
    2236       10717 :     case al_CYCLIC: return algalgmultable_cyc(al, x);
    2237        1015 :     case al_CSA: return algalgmultable_csa(al, x);
    2238             :   }
    2239             :   return NULL; /*LCOV_EXCL_LINE*/
    2240             : }
    2241             : 
    2242             : /* on the natural basis */
    2243             : /* assumes x in algebraic form */
    2244             : static GEN
    2245        8960 : algZmultable(GEN al, GEN x) {
    2246        8960 :   pari_sp av = avma;
    2247        8960 :   GEN res = NULL, x0;
    2248        8960 :   long tx = alg_model(al,x);
    2249        8960 :   switch(tx) {
    2250           0 :     case al_TRIVIAL:
    2251           0 :       x0 = gel(x,1);
    2252           0 :       if (typ(x0)==t_POLMOD) x0 = gel(x0,2);
    2253           0 :       if (typ(x0)==t_POL) x0 = constant_coeff(x0);
    2254           0 :       res = mkmatcopy(mkcol(x0));
    2255           0 :       break;
    2256        8960 :     case al_ALGEBRAIC: res = algmtK2Z(al,algalgmultable(al,x)); break;
    2257             :   }
    2258        8960 :   return gerepileupto(av,res);
    2259             : }
    2260             : 
    2261             : /* x integral */
    2262             : static GEN
    2263       36561 : algbasisrightmultable(GEN al, GEN x)
    2264             : {
    2265       36561 :   long N = alg_get_absdim(al), i,j,k;
    2266       36561 :   GEN res = zeromatcopy(N,N), c, mt = alg_get_multable(al), p = alg_get_char(al);
    2267       36561 :   if (gequal0(p)) p = NULL;
    2268      330862 :   for (i=1; i<=N; i++) {
    2269      294301 :     c = gel(x,i);
    2270      294301 :     if (!gequal0(c)) {
    2271      872200 :       for (j=1; j<=N; j++)
    2272     7417690 :       for(k=1; k<=N; k++) {
    2273     6639682 :         if (p) gcoeff(res,k,j) = Fp_add(gcoeff(res,k,j), Fp_mul(c, gcoeff(gel(mt,j),k,i), p), p);
    2274     5014814 :         else gcoeff(res,k,j) = addii(gcoeff(res,k,j), mulii(c, gcoeff(gel(mt,j),k,i)));
    2275             :       }
    2276             :     }
    2277             :   }
    2278       36561 :   return res;
    2279             : }
    2280             : 
    2281             : /* basis for matrices : 1, E_{i,j} for (i,j)!=(1,1) */
    2282             : /* index : ijk = ((i-1)*N+j-1)*n + k */
    2283             : /* square matrices only, coefficients in basis form, shallow function */
    2284             : static GEN
    2285       20097 : algmat2basis(GEN al, GEN M)
    2286             : {
    2287       20097 :   long n = alg_get_absdim(al), N = lg(M)-1, i, j, k, ij, ijk;
    2288             :   GEN res, x;
    2289       20097 :   res = zerocol(N*N*n);
    2290       60291 :   for (i=1; i<=N; i++) {
    2291      120582 :     for (j=1, ij=(i-1)*N+1; j<=N; j++, ij++) {
    2292       80388 :       x = gcoeff(M,i,j);
    2293      660772 :       for (k=1, ijk=(ij-1)*n+1; k<=n; k++, ijk++) {
    2294      580384 :         gel(res, ijk) = gel(x, k);
    2295      580384 :         if (i>1 && i==j) gel(res, ijk) = gsub(gel(res,ijk), gel(res,k));
    2296             :       }
    2297             :     }
    2298             :   }
    2299             : 
    2300       20097 :   return res;
    2301             : }
    2302             : 
    2303             : static GEN
    2304         294 : algbasis2mat(GEN al, GEN M, long N)
    2305             : {
    2306         294 :   long n = alg_get_absdim(al), i, j, k, ij, ijk;
    2307             :   GEN res, x;
    2308         294 :   res = zeromatcopy(N,N);
    2309         882 :   for (i=1; i<=N; i++)
    2310        1764 :   for (j=1; j<=N; j++)
    2311        1176 :     gcoeff(res,i,j) = zerocol(n);
    2312             : 
    2313         882 :   for (i=1; i<=N; i++) {
    2314        1764 :     for (j=1, ij=(i-1)*N+1; j<=N; j++, ij++) {
    2315        1176 :       x = gcoeff(res,i,j);
    2316        9240 :       for (k=1, ijk=(ij-1)*n+1; k<=n; k++, ijk++) {
    2317        8064 :         gel(x,k) = gel(M,ijk);
    2318        8064 :         if (i>1 && i==j) gel(x,k) = gadd(gel(x,k), gel(M,k));
    2319             :       }
    2320             :     }
    2321             :   }
    2322             : 
    2323         294 :   return res;
    2324             : }
    2325             : 
    2326             : static GEN
    2327       20020 : algmatbasis_ei(GEN al, long ijk, long N)
    2328             : {
    2329       20020 :   long n = alg_get_absdim(al), i, j, k, ij;
    2330             :   GEN res;
    2331             : 
    2332       20020 :   res = zeromatcopy(N,N);
    2333       60060 :   for (i=1; i<=N; i++)
    2334      120120 :   for (j=1; j<=N; j++)
    2335       80080 :     gcoeff(res,i,j) = zerocol(n);
    2336             : 
    2337       20020 :   k = ijk%n;
    2338       20020 :   if (k==0) k=n;
    2339       20020 :   ij = (ijk-k)/n+1;
    2340             : 
    2341       20020 :   if (ij==1) {
    2342       15015 :     for (i=1; i<=N; i++)
    2343       10010 :       gcoeff(res,i,i) = col_ei(n,k);
    2344        5005 :     return res;
    2345             :   }
    2346             : 
    2347       15015 :   j = ij%N;
    2348       15015 :   if (j==0) j=N;
    2349       15015 :   i = (ij-j)/N+1;
    2350             : 
    2351       15015 :   gcoeff(res,i,j) = col_ei(n,k);
    2352       15015 :   return res;
    2353             : }
    2354             : 
    2355             : /* FIXME lazy implementation! */
    2356             : static GEN
    2357         777 : algleftmultable_mat(GEN al, GEN M)
    2358             : {
    2359         777 :   long N = lg(M)-1, n = alg_get_absdim(al), D = N*N*n, j;
    2360             :   GEN res, x, Mx;
    2361         777 :   if (N == 0) return cgetg(1, t_MAT);
    2362         770 :   if (N != nbrows(M)) pari_err_DIM("algleftmultable_mat (nonsquare)");
    2363         749 :   res = cgetg(D+1, t_MAT);
    2364       20769 :   for (j=1; j<=D; j++) {
    2365       20020 :     x = algmatbasis_ei(al, j, N);
    2366       20020 :     Mx = algmul(al, M, x);
    2367       20020 :     gel(res, j) = algmat2basis(al, Mx);
    2368             :   }
    2369         749 :   return res;
    2370             : }
    2371             : 
    2372             : /* left multiplication table on integral basis */
    2373             : static GEN
    2374        6951 : algleftmultable(GEN al, GEN x)
    2375             : {
    2376        6951 :   pari_sp av = avma;
    2377             :   long tx;
    2378             :   GEN res;
    2379             : 
    2380        6951 :   checkalg(al);
    2381        6951 :   tx = alg_model(al,x);
    2382        6944 :   switch(tx) {
    2383          98 :     case al_TRIVIAL : res = mkmatcopy(mkcol(gel(x,1))); break;
    2384         196 :     case al_ALGEBRAIC : x = algalgtobasis(al,x);
    2385        6328 :     case al_BASIS : res = algbasismultable(al,x); break;
    2386         518 :     case al_MATRIX : res = algleftmultable_mat(al,x); break;
    2387             :     default : return NULL; /* LCOV_EXCL_LINE */
    2388             :   }
    2389        6937 :   return gerepileupto(av,res);
    2390             : }
    2391             : 
    2392             : static GEN
    2393        4102 : algbasissplittingmatrix_csa(GEN al, GEN x)
    2394             : {
    2395        4102 :   long d = alg_get_degree(al), i, j;
    2396        4102 :   GEN rnf = alg_get_splittingfield(al), splba = alg_get_splittingbasis(al), splbainv = alg_get_splittingbasisinv(al), M;
    2397        4102 :   M = algbasismultable(al,x);
    2398        4102 :   M = RgM_mul(M, splba); /* TODO best order ? big matrix /Q vs small matrix /nf */
    2399        4102 :   M = RgM_mul(splbainv, M);
    2400       12131 :   for (i=1; i<=d; i++)
    2401       23912 :   for (j=1; j<=d; j++)
    2402       15883 :     gcoeff(M,i,j) = rnfeltabstorel(rnf, gcoeff(M,i,j));
    2403        4102 :   return M;
    2404             : }
    2405             : 
    2406             : GEN
    2407        7399 : algtomatrix(GEN al, GEN x, long abs)
    2408             : {
    2409        7399 :   pari_sp av = avma;
    2410        7399 :   GEN res = NULL;
    2411             :   long ta, tx, i, j;
    2412        7399 :   checkalg(al);
    2413        7399 :   ta = alg_type(al);
    2414        7399 :   if (abs || ta==al_TABLE) return algleftmultable(al,x);
    2415        6622 :   tx = alg_model(al,x);
    2416        6622 :   if (tx==al_MATRIX) {
    2417         469 :     if (lg(x) == 1) return cgetg(1, t_MAT);
    2418         441 :     res = zeromatcopy(nbrows(x),lg(x)-1);
    2419        1323 :     for (j=1; j<lg(x); j++)
    2420        2618 :     for (i=1; i<lgcols(x); i++)
    2421        1736 :       gcoeff(res,i,j) = algtomatrix(al,gcoeff(x,i,j),0);
    2422         441 :     res = shallowmatconcat(res);
    2423             :   }
    2424        6153 :   else switch(alg_type(al))
    2425             :   {
    2426        2051 :     case al_CYCLIC:
    2427        2051 :       if (tx==al_BASIS) x = algbasistoalg(al,x);
    2428        2051 :       res = algalgmultable(al,x);
    2429        2051 :       break;
    2430        4102 :     case al_CSA:
    2431        4102 :       if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
    2432        4102 :       res = algbasissplittingmatrix_csa(al,x);
    2433        4102 :       break;
    2434           0 :     default:
    2435           0 :       pari_err_DOMAIN("algtomatrix", "alg_type(al)", "=", stoi(alg_type(al)), stoi(alg_type(al)));
    2436             :   }
    2437        6594 :   return gerepilecopy(av,res);
    2438             : }
    2439             : 
    2440             : /*  x^(-1)*y, NULL if no solution */
    2441             : static GEN
    2442        1715 : algdivl_i(GEN al, GEN x, GEN y, long tx, long ty) {
    2443        1715 :   pari_sp av = avma;
    2444        1715 :   GEN res, p = alg_get_char(al), mtx;
    2445        1715 :   if (tx != ty) {
    2446         343 :     if (tx==al_ALGEBRAIC) { x = algalgtobasis(al,x); tx=al_BASIS; }
    2447         343 :     if (ty==al_ALGEBRAIC) { y = algalgtobasis(al,y); ty=al_BASIS; }
    2448             :   }
    2449        1715 :   if (ty == al_MATRIX)
    2450             :   {
    2451          77 :     if (alg_type(al) != al_TABLE) y = algalgtobasis(al,y);
    2452          77 :     y = algmat2basis(al,y);
    2453             :   }
    2454        1715 :   if (signe(p)) res = FpM_FpC_invimage(algbasismultable(al,x),y,p);
    2455             :   else
    2456             :   {
    2457        1526 :     if (ty==al_ALGEBRAIC)   mtx = algalgmultable(al,x);
    2458         819 :     else                    mtx = algleftmultable(al,x);
    2459        1526 :     res = inverseimage(mtx,y);
    2460             :   }
    2461        1715 :   if (!res || lg(res)==1) return gc_NULL(av);
    2462        1687 :   if (tx == al_MATRIX) {
    2463         294 :     res = algbasis2mat(al, res, lg(x)-1);
    2464         294 :     return gerepilecopy(av,res);
    2465             :   }
    2466        1393 :   return gerepileupto(av,res);
    2467             : }
    2468             : static GEN
    2469         721 : algdivl_i2(GEN al, GEN x, GEN y)
    2470             : {
    2471             :   long tx, ty;
    2472         721 :   checkalg(al);
    2473         721 :   tx = alg_model(al,x);
    2474         714 :   ty = alg_model(al,y);
    2475         714 :   if (tx == al_MATRIX) {
    2476         119 :     if (ty != al_MATRIX) pari_err_TYPE2("\\", x, y);
    2477         112 :     if (lg(y) == 1) return cgetg(1, t_MAT);
    2478         105 :     if (lg(x) == 1) return NULL;
    2479          98 :     if (lgcols(x) != lgcols(y)) pari_err_DIM("algdivl");
    2480          91 :     if (lg(x) != lgcols(x) || lg(y) != lgcols(y))
    2481          14 :       pari_err_DIM("algdivl (nonsquare)");
    2482             :   }
    2483         672 :   return algdivl_i(al,x,y,tx,ty);
    2484             : }
    2485             : 
    2486         672 : GEN algdivl(GEN al, GEN x, GEN y)
    2487             : {
    2488             :   GEN z;
    2489         672 :   z = algdivl_i2(al,x,y);
    2490         637 :   if (!z) pari_err_INV("algdivl", x);
    2491         623 :   return z;
    2492             : }
    2493             : 
    2494             : int
    2495          49 : algisdivl(GEN al, GEN x, GEN y, GEN* ptz)
    2496             : {
    2497          49 :   pari_sp av = avma;
    2498          49 :   GEN z = algdivl_i2(al,x,y);
    2499          49 :   if (!z) return gc_bool(av,0);
    2500          42 :   if (ptz != NULL) *ptz = z;
    2501          42 :   return 1;
    2502             : }
    2503             : 
    2504             : static GEN
    2505        1148 : alginv_i(GEN al, GEN x)
    2506             : {
    2507        1148 :   pari_sp av = avma;
    2508        1148 :   GEN res = NULL, p = alg_get_char(al);
    2509        1148 :   long tx = alg_model(al,x), n;
    2510        1127 :   switch(tx) {
    2511          63 :     case al_TRIVIAL :
    2512          63 :       if (signe(p)) { res = mkcol(Fp_inv(gel(x,1),p)); break; }
    2513          49 :       else          { res = mkcol(ginv(gel(x,1))); break; }
    2514         455 :     case al_ALGEBRAIC :
    2515         455 :       switch(alg_type(al)) {
    2516         350 :         case al_CYCLIC: n = alg_get_degree(al); break;
    2517         105 :         case al_CSA: n = alg_get_dim(al); break;
    2518             :         default: return NULL; /* LCOV_EXCL_LINE */
    2519             :       }
    2520         455 :       res = algdivl_i(al, x, col_ei(n,1), tx, al_ALGEBRAIC); break;
    2521         371 :     case al_BASIS : res = algdivl_i(al, x, col_ei(alg_get_absdim(al),1), tx,
    2522         371 :                                                             al_BASIS); break;
    2523         238 :     case al_MATRIX :
    2524         238 :       n = lg(x)-1;
    2525         238 :       if (n==0) return cgetg(1, t_MAT);
    2526         224 :       if (n != nbrows(x)) pari_err_DIM("alginv_i (nonsquare)");
    2527         217 :       res = algdivl_i(al, x, col_ei(n*n*alg_get_absdim(al),1), tx, al_BASIS);
    2528             :         /* cheat on type because wrong dimension */
    2529             :   }
    2530        1106 :   if (!res) return gc_NULL(av);
    2531        1092 :   return gerepilecopy(av,res);
    2532             : }
    2533             : GEN
    2534        1078 : alginv(GEN al, GEN x)
    2535             : {
    2536             :   GEN z;
    2537        1078 :   checkalg(al);
    2538        1078 :   z = alginv_i(al,x);
    2539        1050 :   if (!z) pari_err_INV("alginv", x);
    2540        1043 :   return z;
    2541             : }
    2542             : 
    2543             : int
    2544          70 : algisinv(GEN al, GEN x, GEN* ptix)
    2545             : {
    2546          70 :   pari_sp av = avma;
    2547             :   GEN ix;
    2548          70 :   checkalg(al);
    2549          70 :   ix = alginv_i(al,x);
    2550          70 :   if (!ix) return gc_bool(av,0);
    2551          63 :   if (ptix != NULL) *ptix = ix;
    2552          63 :   return 1;
    2553             : }
    2554             : 
    2555             : /*  x*y^(-1)  */
    2556             : GEN
    2557         406 : algdivr(GEN al, GEN x, GEN y) { return algmul(al, x, alginv(al, y)); }
    2558             : 
    2559       25214 : static GEN _mul(void* data, GEN x, GEN y) { return algmul((GEN)data,x,y); }
    2560       47712 : static GEN _sqr(void* data, GEN x) { return algsqr((GEN)data,x); }
    2561             : 
    2562             : static GEN
    2563          21 : algmatid(GEN al, long N)
    2564             : {
    2565          21 :   long n = alg_get_absdim(al), i, j;
    2566             :   GEN res, one, zero;
    2567             : 
    2568          21 :   res = zeromatcopy(N,N);
    2569          21 :   one = col_ei(n,1);
    2570          21 :   zero = zerocol(n);
    2571          49 :   for (i=1; i<=N; i++)
    2572          84 :   for (j=1; j<=N; j++)
    2573          56 :     gcoeff(res,i,j) = i==j ? one : zero;
    2574          21 :   return res;
    2575             : }
    2576             : 
    2577             : GEN
    2578       12222 : algpow(GEN al, GEN x, GEN n)
    2579             : {
    2580       12222 :   pari_sp av = avma;
    2581             :   GEN res;
    2582       12222 :   checkalg(al);
    2583       12222 :   switch(signe(n)) {
    2584          28 :     case 0:
    2585          28 :       if (alg_model(al,x) == al_MATRIX)
    2586          21 :         res = algmatid(al,lg(x)-1);
    2587             :       else
    2588           7 :         res = col_ei(alg_get_absdim(al),1);
    2589          28 :       return res;
    2590       12110 :     case 1:
    2591       12110 :       res = gen_pow_i(x, n, (void*)al, _sqr, _mul); break;
    2592          84 :     default: /* -1 */
    2593          84 :       res = gen_pow_i(alginv(al,x), gneg(n), (void*)al, _sqr, _mul);
    2594             :   }
    2595       12187 :   return gerepilecopy(av,res);
    2596             : }
    2597             : 
    2598             : static GEN
    2599         378 : algredcharpoly_i(GEN al, GEN x, long v)
    2600             : {
    2601         378 :   GEN rnf = alg_get_splittingfield(al);
    2602         378 :   GEN cp = charpoly(algtomatrix(al,x,0),v);
    2603         371 :   long i, m = lg(cp);
    2604        1540 :   for (i=2; i<m; i++) gel(cp,i) = rnfeltdown(rnf, gel(cp,i));
    2605         371 :   return cp;
    2606             : }
    2607             : 
    2608             : /* assumes al is CSA or CYCLIC */
    2609             : static GEN
    2610         385 : algredcharpoly(GEN al, GEN x, long v)
    2611             : {
    2612         385 :   pari_sp av = avma;
    2613         385 :   long w = gvar(rnf_get_pol(alg_get_center(al)));
    2614         385 :   if (varncmp(v,w)>=0) pari_err_PRIORITY("algredcharpoly",pol_x(v),">=",w);
    2615         378 :   switch(alg_type(al))
    2616             :   {
    2617         378 :     case al_CYCLIC:
    2618             :     case al_CSA:
    2619         378 :       return gerepileupto(av, algredcharpoly_i(al, x, v));
    2620             :   }
    2621             :   return NULL; /*LCOV_EXCL_LINE*/
    2622             : }
    2623             : 
    2624             : static GEN
    2625       20928 : algbasischarpoly(GEN al, GEN x, long v)
    2626             : {
    2627       20928 :   pari_sp av = avma;
    2628       20928 :   GEN p = alg_get_char(al), mx;
    2629       20928 :   if (alg_model(al,x) == al_MATRIX) mx = algleftmultable_mat(al,x);
    2630       20837 :   else                              mx = algbasismultable(al,x);
    2631       20921 :   if (signe(p)) {
    2632       19017 :     GEN res = FpM_charpoly(mx,p);
    2633       19017 :     setvarn(res,v);
    2634       19017 :     return gerepileupto(av, res);
    2635             :   }
    2636        1904 :   return gerepileupto(av, charpoly(mx,v));
    2637             : }
    2638             : 
    2639             : GEN
    2640       20998 : algcharpoly(GEN al, GEN x, long v, long abs)
    2641             : {
    2642       20998 :   checkalg(al);
    2643       20998 :   if (v<0) v=0;
    2644             : 
    2645             :   /* gneg(x[1]) left on stack */
    2646       20998 :   if (alg_model(al,x) == al_TRIVIAL) {
    2647          56 :     GEN p = alg_get_char(al);
    2648          56 :     if (signe(p)) return deg1pol(gen_1,Fp_neg(gel(x,1),p),v);
    2649          42 :     return deg1pol(gen_1,gneg(gel(x,1)),v);
    2650             :   }
    2651             : 
    2652       20935 :   switch(alg_type(al)) {
    2653         490 :     case al_CYCLIC: case al_CSA:
    2654         490 :       if (abs)
    2655             :       {
    2656         105 :         if (alg_model(al,x)==al_ALGEBRAIC) x = algalgtobasis(al,x);
    2657             :       }
    2658         385 :       else return algredcharpoly(al,x,v);
    2659       20550 :     case al_TABLE: return algbasischarpoly(al,x,v);
    2660             :     default : return NULL; /* LCOV_EXCL_LINE */
    2661             :   }
    2662             : }
    2663             : 
    2664             : /* assumes x in basis form */
    2665             : static GEN
    2666      236565 : algabstrace(GEN al, GEN x)
    2667             : {
    2668      236565 :   pari_sp av = avma;
    2669      236565 :   GEN res = NULL, p = alg_get_char(al);
    2670      236565 :   if (signe(p)) return FpV_dotproduct(x, alg_get_tracebasis(al), p);
    2671       42644 :   switch(alg_model(al,x)) {
    2672          84 :     case al_TRIVIAL: return gcopy(gel(x,1)); break;
    2673       42560 :     case al_BASIS: res = RgV_dotproduct(x, alg_get_tracebasis(al)); break;
    2674             :   }
    2675       42560 :   return gerepileupto(av,res);
    2676             : }
    2677             : 
    2678             : static GEN
    2679        1372 : algredtrace(GEN al, GEN x)
    2680             : {
    2681        1372 :   pari_sp av = avma;
    2682        1372 :   GEN res = NULL;
    2683        1372 :   switch(alg_model(al,x)) {
    2684          35 :     case al_TRIVIAL: return gcopy(gel(x,1)); break;
    2685         490 :     case al_BASIS: return algredtrace(al, algbasistoalg(al,x));
    2686             :                    /* TODO precompute too? */
    2687         847 :     case al_ALGEBRAIC:
    2688         847 :       switch(alg_type(al))
    2689             :       {
    2690         553 :         case al_CYCLIC:
    2691         553 :           res = rnfelttrace(alg_get_splittingfield(al),gel(x,1));
    2692         553 :           break;
    2693         294 :         case al_CSA:
    2694         294 :           res = gtrace(algalgmultable_csa(al,x));
    2695         294 :           res = gdiv(res, stoi(alg_get_degree(al)));
    2696         294 :           break;
    2697             :         default: return NULL; /* LCOV_EXCL_LINE */
    2698             :       }
    2699         847 :   }
    2700         847 :   return gerepileupto(av,res);
    2701             : }
    2702             : 
    2703             : static GEN
    2704         308 : algtrace_mat(GEN al, GEN M, long abs) {
    2705         308 :   pari_sp av = avma;
    2706         308 :   long N = lg(M)-1, i;
    2707         308 :   GEN res, p = alg_get_char(al);
    2708         308 :   if (N == 0) return gen_0;
    2709         294 :   if (N != nbrows(M)) pari_err_DIM("algtrace_mat (nonsquare)");
    2710             : 
    2711         287 :   if (!signe(p)) p = NULL;
    2712         287 :   res = algtrace(al, gcoeff(M,1,1), abs);
    2713         574 :   for (i=2; i<=N; i++) {
    2714         287 :     if (p)  res = Fp_add(res, algtrace(al,gcoeff(M,i,i),abs), p);
    2715         280 :     else    res = gadd(res, algtrace(al,gcoeff(M,i,i),abs));
    2716             :   }
    2717         287 :   if (abs || alg_type(al) == al_TABLE) res = gmulgs(res, N); /* absolute trace */
    2718         287 :   return gerepileupto(av, res);
    2719             : }
    2720             : 
    2721             : GEN
    2722        1519 : algtrace(GEN al, GEN x, long abs)
    2723             : {
    2724        1519 :   checkalg(al);
    2725        1519 :   if (alg_model(al,x) == al_MATRIX) return algtrace_mat(al,x,abs);
    2726        1211 :   switch(alg_type(al)) {
    2727        1078 :     case al_CYCLIC: case al_CSA:
    2728        1078 :       if (!abs) return algredtrace(al,x);
    2729         196 :       if (alg_model(al,x)==al_ALGEBRAIC) x = algalgtobasis(al,x);
    2730         329 :     case al_TABLE: return algabstrace(al,x);
    2731             :     default : return NULL; /* LCOV_EXCL_LINE */
    2732             :   }
    2733             : }
    2734             : 
    2735             : static GEN
    2736       40705 : ZM_trace(GEN x)
    2737             : {
    2738       40705 :   long i, lx = lg(x);
    2739             :   GEN t;
    2740       40705 :   if (lx < 3) return lx == 1? gen_0: gcopy(gcoeff(x,1,1));
    2741       39907 :   t = gcoeff(x,1,1);
    2742      680029 :   for (i = 2; i < lx; i++) t = addii(t, gcoeff(x,i,i));
    2743       39907 :   return t;
    2744             : }
    2745             : static GEN
    2746      126810 : FpM_trace(GEN x, GEN p)
    2747             : {
    2748      126810 :   long i, lx = lg(x);
    2749             :   GEN t;
    2750      126810 :   if (lx < 3) return lx == 1? gen_0: gcopy(gcoeff(x,1,1));
    2751      119096 :   t = gcoeff(x,1,1);
    2752      888498 :   for (i = 2; i < lx; i++) t = Fp_add(t, gcoeff(x,i,i), p);
    2753      119096 :   return t;
    2754             : }
    2755             : 
    2756             : static GEN
    2757       38864 : algtracebasis(GEN al)
    2758             : {
    2759       38864 :   pari_sp av = avma;
    2760       38864 :   GEN mt = alg_get_multable(al), p = alg_get_char(al);
    2761       38864 :   long i, l = lg(mt);
    2762       38864 :   GEN v = cgetg(l, t_VEC);
    2763      165674 :   if (signe(p)) for (i=1; i < l; i++) gel(v,i) = FpM_trace(gel(mt,i), p);
    2764       46130 :   else          for (i=1; i < l; i++) gel(v,i) = ZM_trace(gel(mt,i));
    2765       38864 :   return gerepileupto(av,v);
    2766             : }
    2767             : 
    2768             : /* Assume: i > 0, expo := p^i <= absdim, x contained in I_{i-1} given by mult
    2769             :  * table modulo modu=p^(i+1). Return Tr(x^(p^i)) mod modu */
    2770             : static ulong
    2771       24472 : algtracei(GEN mt, ulong p, ulong expo, ulong modu)
    2772             : {
    2773       24472 :   pari_sp av = avma;
    2774       24472 :   long j, l = lg(mt);
    2775       24472 :   ulong tr = 0;
    2776       24472 :   mt = Flm_powu(mt,expo,modu);
    2777      267995 :   for (j=1; j<l; j++) tr += ucoeff(mt,j,j);
    2778       24472 :   return gc_ulong(av, (tr/expo) % p);
    2779             : }
    2780             : 
    2781             : GEN
    2782         952 : algnorm(GEN al, GEN x, long abs)
    2783             : {
    2784         952 :   pari_sp av = avma;
    2785             :   long tx;
    2786             :   GEN p, rnf, res, mx;
    2787         952 :   checkalg(al);
    2788         952 :   p = alg_get_char(al);
    2789         952 :   tx = alg_model(al,x);
    2790         952 :   if (signe(p)) {
    2791          21 :     if (tx == al_MATRIX)    mx = algleftmultable_mat(al,x);
    2792          14 :     else                    mx = algbasismultable(al,x);
    2793          21 :     return gerepileupto(av, FpM_det(mx,p));
    2794             :   }
    2795         931 :   if (tx == al_TRIVIAL) return gcopy(gel(x,1));
    2796             : 
    2797         889 :   switch(alg_type(al)) {
    2798         819 :     case al_CYCLIC: case al_CSA:
    2799         819 :       if (abs)
    2800             :       {
    2801         196 :         if (alg_model(al,x)==al_ALGEBRAIC) x = algalgtobasis(al,x);
    2802             :       }
    2803             :       else
    2804             :       {
    2805         623 :         rnf = alg_get_splittingfield(al);
    2806         623 :         res = rnfeltdown(rnf, det(algtomatrix(al,x,0)));
    2807         616 :         break;
    2808             :       }
    2809             :     case al_TABLE:
    2810         266 :       if (tx == al_MATRIX)  mx = algleftmultable_mat(al,x);
    2811         105 :       else                  mx = algbasismultable(al,x);
    2812         259 :       res = det(mx);
    2813         259 :       break;
    2814             :     default: return NULL; /* LCOV_EXCL_LINE */
    2815             :   }
    2816         875 :   return gerepileupto(av, res);
    2817             : }
    2818             : 
    2819             : static GEN
    2820       48384 : algalgtonat_cyc(GEN al, GEN x)
    2821             : {
    2822       48384 :   pari_sp av = avma;
    2823       48384 :   GEN nf = alg_get_abssplitting(al), rnf = alg_get_splittingfield(al), res, c;
    2824       48384 :   long n = alg_get_degree(al), N = nf_get_degree(nf), i, i1;
    2825       48384 :   res = zerocol(N*n);
    2826      148358 :   for (i=0; i<n; i++) {
    2827       99974 :     c = gel(x,i+1);
    2828       99974 :     c = rnfeltreltoabs(rnf,c);
    2829       99974 :     if (!gequal0(c)) {
    2830       76006 :       c = algtobasis(nf,c);
    2831      413084 :       for (i1=1; i1<=N; i1++) gel(res,i*N+i1) = gel(c,i1);
    2832             :     }
    2833             :   }
    2834       48384 :   return gerepilecopy(av, res);
    2835             : }
    2836             : 
    2837             : static GEN
    2838       11256 : algalgtonat_csa(GEN al, GEN x)
    2839             : {
    2840       11256 :   pari_sp av = avma;
    2841       11256 :   GEN nf = alg_get_center(al), res, c;
    2842       11256 :   long d2 = alg_get_dim(al), n = nf_get_degree(nf), i, i1;
    2843       11256 :   res = zerocol(d2*n);
    2844       56133 :   for (i=0; i<d2; i++) {
    2845       44877 :     c = gel(x,i+1);
    2846       44877 :     if (!gequal0(c)) {
    2847       31318 :       c = algtobasis(nf,c);
    2848       94395 :       for (i1=1; i1<=n; i1++) gel(res,i*n+i1) = gel(c,i1);
    2849             :     }
    2850             :   }
    2851       11256 :   return gerepilecopy(av, res);
    2852             : }
    2853             : 
    2854             : /* assumes al CSA or CYCLIC */
    2855             : static GEN
    2856       59640 : algalgtonat(GEN al, GEN x)
    2857             : {
    2858       59640 :   switch(alg_type(al))
    2859             :   {
    2860       48384 :     case al_CYCLIC: return algalgtonat_cyc(al, x);
    2861       11256 :     case al_CSA: return algalgtonat_csa(al, x);
    2862             :   }
    2863             :   return NULL; /*LCOV_EXCL_LINE*/
    2864             : }
    2865             : 
    2866             : static GEN
    2867       10381 : algnattoalg_cyc(GEN al, GEN x)
    2868             : {
    2869       10381 :   pari_sp av = avma;
    2870       10381 :   GEN nf = alg_get_abssplitting(al), rnf = alg_get_splittingfield(al), res, c;
    2871       10381 :   long n = alg_get_degree(al), N = nf_get_degree(nf), i, i1;
    2872       10381 :   res = zerocol(n);
    2873       10381 :   c = zerocol(N);
    2874       44926 :   for (i=0; i<n; i++) {
    2875      292978 :     for (i1=1; i1<=N; i1++) gel(c,i1) = gel(x,i*N+i1);
    2876       34545 :     gel(res,i+1) = rnfeltabstorel(rnf,basistoalg(nf,c));
    2877             :   }
    2878       10381 :   return gerepilecopy(av, res);
    2879             : }
    2880             : 
    2881             : static GEN
    2882        1225 : algnattoalg_csa(GEN al, GEN x)
    2883             : {
    2884        1225 :   pari_sp av = avma;
    2885        1225 :   GEN nf = alg_get_center(al), res, c;
    2886        1225 :   long d2 = alg_get_dim(al), n = nf_get_degree(nf), i, i1;
    2887        1225 :   res = zerocol(d2);
    2888        1225 :   c = zerocol(n);
    2889        6608 :   for (i=0; i<d2; i++) {
    2890       18494 :     for (i1=1; i1<=n; i1++) gel(c,i1) = gel(x,i*n+i1);
    2891        5383 :     gel(res,i+1) = basistoalg(nf,c);
    2892             :   }
    2893        1225 :   return gerepilecopy(av, res);
    2894             : }
    2895             : 
    2896             : /* assumes al CSA or CYCLIC */
    2897             : static GEN
    2898       11606 : algnattoalg(GEN al, GEN x)
    2899             : {
    2900       11606 :   switch(alg_type(al))
    2901             :   {
    2902       10381 :     case al_CYCLIC: return algnattoalg_cyc(al, x);
    2903        1225 :     case al_CSA: return algnattoalg_csa(al, x);
    2904             :   }
    2905             :   return NULL; /*LCOV_EXCL_LINE*/
    2906             : }
    2907             : 
    2908             : static GEN
    2909         182 : algalgtobasis_mat(GEN al, GEN x) /* componentwise */
    2910             : {
    2911         182 :   pari_sp av = avma;
    2912             :   long lx, lxj, i, j;
    2913             :   GEN res;
    2914         182 :   lx = lg(x);
    2915         182 :   res = cgetg(lx, t_MAT);
    2916         546 :   for (j=1; j<lx; j++) {
    2917         364 :     lxj = lg(gel(x,j));
    2918         364 :     gel(res,j) = cgetg(lxj, t_COL);
    2919        1092 :     for (i=1; i<lxj; i++)
    2920         728 :       gcoeff(res,i,j) = algalgtobasis(al,gcoeff(x,i,j));
    2921             :   }
    2922         182 :   return gerepilecopy(av,res);
    2923             : }
    2924             : GEN
    2925       60095 : algalgtobasis(GEN al, GEN x)
    2926             : {
    2927             :   pari_sp av;
    2928             :   long tx;
    2929       60095 :   checkalg(al);
    2930       60095 :   if (alg_type(al) == al_TABLE) pari_err_TYPE("algalgtobasis [use alginit]", al);
    2931       60081 :   tx = alg_model(al,x);
    2932       60081 :   if (tx==al_BASIS) return gcopy(x);
    2933       59822 :   if (tx==al_MATRIX) return algalgtobasis_mat(al,x);
    2934       59640 :   av = avma;
    2935       59640 :   x = algalgtonat(al,x);
    2936       59640 :   x = RgM_RgC_mul(alg_get_invbasis(al),x);
    2937       59640 :   return gerepileupto(av, x);
    2938             : }
    2939             : 
    2940             : static GEN
    2941         119 : algbasistoalg_mat(GEN al, GEN x) /* componentwise */
    2942             : {
    2943         119 :   long j, lx = lg(x);
    2944         119 :   GEN res = cgetg(lx, t_MAT);
    2945         357 :   for (j=1; j<lx; j++) {
    2946         238 :     long i, lxj = lg(gel(x,j));
    2947         238 :     gel(res,j) = cgetg(lxj, t_COL);
    2948         714 :     for (i=1; i<lxj; i++) gcoeff(res,i,j) = algbasistoalg(al,gcoeff(x,i,j));
    2949             :   }
    2950         119 :   return res;
    2951             : }
    2952             : GEN
    2953        2912 : algbasistoalg(GEN al, GEN x)
    2954             : {
    2955             :   pari_sp av;
    2956             :   long tx;
    2957        2912 :   checkalg(al);
    2958        2912 :   if (alg_type(al) == al_TABLE) pari_err_TYPE("algbasistoalg [use alginit]", al);
    2959        2898 :   tx = alg_model(al,x);
    2960        2898 :   if (tx==al_ALGEBRAIC) return gcopy(x);
    2961        2765 :   if (tx==al_MATRIX) return algbasistoalg_mat(al,x);
    2962        2646 :   av = avma;
    2963        2646 :   x = RgM_RgC_mul(alg_get_basis(al),x);
    2964        2646 :   x = algnattoalg(al,x);
    2965        2646 :   return gerepileupto(av, x);
    2966             : }
    2967             : 
    2968             : GEN
    2969       18305 : algrandom(GEN al, GEN b)
    2970             : {
    2971             :   GEN res, p, N;
    2972             :   long i, n;
    2973       18305 :   if (typ(b) != t_INT) pari_err_TYPE("algrandom",b);
    2974       18298 :   if (signe(b)<0) pari_err_DOMAIN("algrandom", "b", "<", gen_0, b);
    2975       18291 :   checkalg(al);
    2976       18284 :   n = alg_get_absdim(al);
    2977       18284 :   N = addiu(shifti(b,1), 1); /* left on stack */
    2978       18284 :   p = alg_get_char(al); if (!signe(p)) p = NULL;
    2979       18284 :   res = cgetg(n+1,t_COL);
    2980      163828 :   for (i=1; i<= n; i++)
    2981             :   {
    2982      145544 :     pari_sp av = avma;
    2983      145544 :     GEN t = subii(randomi(N),b);
    2984      145544 :     if (p) t = modii(t, p);
    2985      145544 :     gel(res,i) = gerepileuptoint(av, t);
    2986             :   }
    2987       18284 :   return res;
    2988             : }
    2989             : 
    2990             : /* Assumes pol has coefficients in the same ring as the COL x; x either
    2991             :  * in basis or algebraic form or [x,mx] where mx is the mult. table of x.
    2992             :  TODO more general version: pol with coeffs in center and x in basis form */
    2993             : GEN
    2994       17080 : algpoleval(GEN al, GEN pol, GEN x)
    2995             : {
    2996       17080 :   pari_sp av = avma;
    2997       17080 :   GEN p, mx = NULL, res;
    2998             :   long i;
    2999       17080 :   checkalg(al);
    3000       17080 :   p = alg_get_char(al);
    3001       17080 :   if (typ(pol) != t_POL) pari_err_TYPE("algpoleval", pol);
    3002       17073 :   if (typ(x) == t_VEC)
    3003             :   {
    3004        6097 :     if (lg(x) != 3) pari_err_TYPE("algpoleval [vector must be of length 2]", x);
    3005        6090 :     mx = gel(x,2);
    3006        6090 :     x = gel(x,1);
    3007        6090 :     if (typ(mx)!=t_MAT || !gequal(x,gel(mx,1)))
    3008          21 :       pari_err_TYPE("algpoleval [mx must be the multiplication table of x]", mx);
    3009             :   }
    3010             :   else
    3011             :   {
    3012       10976 :     switch(alg_model(al,x))
    3013             :     {
    3014          14 :       case al_ALGEBRAIC: mx = algalgmultable(al,x); break;
    3015       10934 :       case al_BASIS: if (!RgX_is_QX(pol))
    3016           7 :         pari_err_IMPL("algpoleval with x in basis form and pol not in Q[x]");
    3017       10941 :       case al_TRIVIAL: mx = algbasismultable(al,x); break;
    3018           7 :       default: pari_err_TYPE("algpoleval", x);
    3019             :     }
    3020             :   }
    3021       17024 :   res = zerocol(lg(mx)-1);
    3022       17024 :   if (signe(p)) {
    3023       63971 :     for (i=lg(pol)-1; i>1; i--)
    3024             :     {
    3025       47745 :       gel(res,1) = Fp_add(gel(res,1), gel(pol,i), p);
    3026       47745 :       if (i>2) res = FpM_FpC_mul(mx, res, p);
    3027             :     }
    3028             :   }
    3029             :   else {
    3030        4746 :     for (i=lg(pol)-1; i>1; i--)
    3031             :     {
    3032        3948 :       gel(res,1) = gadd(gel(res,1), gel(pol,i));
    3033        3948 :       if (i>2) res = RgM_RgC_mul(mx, res);
    3034             :     }
    3035             :   }
    3036       17024 :   return gerepileupto(av, res);
    3037             : }
    3038             : 
    3039             : /** GRUNWALD-WANG **/
    3040             : /*
    3041             : Song Wang's PhD thesis (pdf pages)
    3042             : p.25 definition of chi_b. K^Ker(chi_b) = K(b^(1/m))
    3043             : p.26 bound on the conductor (also Cohen adv. GTM 193 p.166)
    3044             : p.21 & p.34 description special case, also on wikipedia:
    3045             : http://en.wikipedia.org/wiki/Grunwald%E2%80%93Wang_theorem#Special_fields
    3046             : p.77 Kummer case
    3047             : */
    3048             : 
    3049             : /* n > 0. Is n = 2^k ? */
    3050             : static int
    3051         154 : uispow2(ulong n) { return !(n &(n-1)); }
    3052             : 
    3053             : static GEN
    3054         175 : get_phi0(GEN bnr, GEN Lpr, GEN Ld, GEN pl, long *pr, long *pn)
    3055             : {
    3056         175 :   const long NTRY = 10; /* FIXME: magic constant */
    3057         175 :   const long n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
    3058         175 :   GEN S = bnr_get_cyc(bnr);
    3059             :   GEN Sst, G, globGmod, loc, X, Rglob, Rloc, H, U, Lconj;
    3060             :   long i, j, r, nbfrob, nbloc, nz, t;
    3061             : 
    3062         175 :   *pn = n;
    3063         175 :   *pr = r = lg(S)-1;
    3064         175 :   if (!r) return NULL;
    3065         154 :   Lconj = NULL;
    3066         154 :   nbloc = nbfrob = lg(Lpr)-1;
    3067         154 :   if (uispow2(n))
    3068             :   {
    3069          84 :     long l = lg(pl), k = 1;
    3070          84 :     GEN real = cgetg(l, t_VECSMALL);
    3071         210 :     for (i=1; i<l; i++)
    3072         126 :       if (pl[i]==-1) real[k++] = i;
    3073          84 :     if (k > 1)
    3074             :     {
    3075          84 :       GEN nf = bnr_get_nf(bnr), I = bid_get_fact(bnr_get_bid(bnr));
    3076          84 :       GEN v, y, C = idealchineseinit(bnr, I);
    3077          84 :       long r1 = nf_get_r1(nf), n = nbrows(I);
    3078          84 :       nbloc += k-1;
    3079          84 :       Lconj = cgetg(k, t_VEC);
    3080          84 :       v = const_vecsmall(r1,1);
    3081          84 :       y = const_vec(n, gen_1);
    3082         210 :       for (i = 1; i < k; i++)
    3083             :       {
    3084         126 :         v[i] = -1; gel(Lconj,i) = idealchinese(nf,mkvec2(C,v),y);
    3085         126 :         v[i] = 1;
    3086             :       }
    3087             :     }
    3088             :   }
    3089             : 
    3090             :   /* compute Z/n-dual */
    3091         154 :   Sst = cgetg(r+1, t_VECSMALL);
    3092         336 :   for (i=1; i<=r; i++) Sst[i] = ugcdiu(gel(S,i), n);
    3093         154 :   if (Sst[1] != n) return NULL;
    3094             : 
    3095         154 :   globGmod = cgetg(r+1,t_MAT);
    3096         154 :   G = cgetg(r+1,t_VECSMALL);
    3097         336 :   for (i=1; i<=r; i++)
    3098             :   {
    3099         182 :     G[i] = n / Sst[i]; /* pairing between S and Sst */
    3100         182 :     gel(globGmod,i) = cgetg(nbloc+1,t_VECSMALL);
    3101             :   }
    3102             : 
    3103             :   /* compute images of Frobenius elements (and complex conjugation) */
    3104         154 :   loc = cgetg(nbloc+1,t_VECSMALL);
    3105         490 :   for (i=1; i<=nbloc; i++) {
    3106             :     long L;
    3107         350 :     if (i<=nbfrob)
    3108             :     {
    3109         224 :       X = gel(Lpr,i);
    3110         224 :       L = Ld[i];
    3111             :     }
    3112             :     else
    3113             :     { /* X = 1 (mod f), sigma_i(x) < 0, positive at all other real places */
    3114         126 :       X = gel(Lconj,i-nbfrob);
    3115         126 :       L = 2;
    3116             :     }
    3117         350 :     X = ZV_to_Flv(isprincipalray(bnr,X), n);
    3118         868 :     for (nz=0,j=1; j<=r; j++)
    3119             :     {
    3120         518 :       ulong c = (X[j] * G[j]) % L;
    3121         518 :       ucoeff(globGmod,i,j) = c;
    3122         518 :       if (c) nz = 1;
    3123             :     }
    3124         350 :     if (!nz) return NULL;
    3125         336 :     loc[i] = L;
    3126             :   }
    3127             : 
    3128             :   /* try some random elements in the dual */
    3129         140 :   Rglob = cgetg(r+1,t_VECSMALL);
    3130         476 :   for (t=0; t<NTRY; t++) {
    3131        1141 :     for (j=1; j<=r; j++) Rglob[j] = random_Fl(Sst[j]);
    3132         469 :     Rloc = zm_zc_mul(globGmod,Rglob);
    3133         952 :     for (i=1; i<=nbloc; i++)
    3134         819 :       if (Rloc[i] % loc[i] == 0) break;
    3135         469 :     if (i > nbloc)
    3136         133 :       return zv_to_ZV(Rglob);
    3137             :   }
    3138             : 
    3139             :   /* try to realize some random elements of the product of the local duals */
    3140           7 :   H = ZM_hnfall_i(shallowconcat(zm_to_ZM(globGmod),
    3141             :                                 diagonal_shallow(zv_to_ZV(loc))), &U, 2);
    3142             :   /* H,U nbloc x nbloc */
    3143           7 :   Rloc = cgetg(nbloc+1,t_COL);
    3144          77 :   for (t=0; t<NTRY; t++) {
    3145             :     /* nonzero random coordinate */ /* TODO add special case ? */
    3146         560 :     for (i=1; i<=nbloc; i++) gel(Rloc,i) = stoi(1 + random_Fl(loc[i]-1));
    3147          70 :     Rglob = hnf_invimage(H, Rloc);
    3148          70 :     if (Rglob)
    3149             :     {
    3150           0 :       Rglob = ZM_ZC_mul(U,Rglob);
    3151           0 :       return vecslice(Rglob,1,r);
    3152             :     }
    3153             :   }
    3154           7 :   return NULL;
    3155             : }
    3156             : 
    3157             : static GEN
    3158         175 : bnrgwsearch(GEN bnr, GEN Lpr, GEN Ld, GEN pl)
    3159             : {
    3160         175 :   pari_sp av = avma;
    3161             :   long n, r;
    3162         175 :   GEN phi0 = get_phi0(bnr,Lpr,Ld,pl, &r,&n), gn, v, H,U;
    3163         175 :   if (!phi0) return gc_const(av, gen_0);
    3164         133 :   gn = stoi(n);
    3165             :   /* compute kernel of phi0 */
    3166         133 :   v = ZV_extgcd(shallowconcat(phi0, gn));
    3167         133 :   U = vecslice(gel(v,2), 1,r);
    3168         133 :   H = ZM_hnfmodid(rowslice(U, 1,r), gn);
    3169         133 :   return gerepileupto(av, H);
    3170             : }
    3171             : 
    3172             : GEN
    3173         133 : bnfgwgeneric(GEN bnf, GEN Lpr, GEN Ld, GEN pl, long var)
    3174             : {
    3175         133 :   pari_sp av = avma;
    3176         133 :   const long n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
    3177             :   forprime_t S;
    3178         133 :   GEN bnr = NULL, ideal = gen_1, nf, dec, H = gen_0, finf, pol;
    3179             :   ulong ell, p;
    3180             :   long deg, i, degell;
    3181         133 :   (void)uisprimepower(n, &ell);
    3182         133 :   nf = bnf_get_nf(bnf);
    3183         133 :   deg = nf_get_degree(nf);
    3184         133 :   degell = ugcd(deg,ell-1);
    3185         133 :   finf = cgetg(lg(pl),t_VEC);
    3186         329 :   for (i=1; i<lg(pl); i++) gel(finf,i) = pl[i]==-1 ? gen_1 : gen_0;
    3187             : 
    3188         133 :   u_forprime_init(&S, 2, ULONG_MAX);
    3189         532 :   while ((p = u_forprime_next(&S))) {
    3190         532 :     if (Fl_powu(p % ell, degell, ell) != 1) continue; /* ell | p^deg-1 ? */
    3191         238 :     dec = idealprimedec(nf, utoipos(p));
    3192         392 :     for (i=1; i<lg(dec); i++) {
    3193         287 :       GEN pp = gel(dec,i);
    3194         287 :       if (RgV_isin(Lpr,pp)) continue;
    3195             :         /* TODO also accept the prime ideals at which there is a condition
    3196             :          * (use local Artin)? */
    3197         231 :       if (smodis(idealnorm(nf,pp),ell) != 1) continue; /* ell | N(pp)-1 ? */
    3198         175 :       ideal = idealmul(bnf,ideal,pp);
    3199             :       /* TODO: give factorization ? */
    3200         175 :       bnr = Buchray(bnf, mkvec2(ideal,finf), nf_INIT);
    3201         175 :       H = bnrgwsearch(bnr,Lpr,Ld,pl);
    3202         175 :       if (H != gen_0)
    3203             :       {
    3204         133 :         pol = rnfkummer(bnr,H,nf_get_prec(nf));
    3205         133 :         setvarn(pol, var);
    3206         133 :         return gerepileupto(av,pol);
    3207             :       }
    3208             :     }
    3209             :   }
    3210             :   pari_err_BUG("bnfgwgeneric (no suitable p)"); /*LCOV_EXCL_LINE*/
    3211             :   return NULL;/*LCOV_EXCL_LINE*/
    3212             : }
    3213             : 
    3214             : /* no garbage collection */
    3215             : static GEN
    3216         245 : localextdeg(GEN nf, GEN pr, GEN cnd, long d, long ell, long n)
    3217             : {
    3218         245 :   long g = n/d;
    3219         245 :   GEN res, modpr, ppr = pr, T, p, gen, k;
    3220         245 :   if (d==1) return gen_1;
    3221         224 :   if (equalsi(ell,pr_get_p(pr))) { /* ell == p */
    3222          21 :     res = nfadd(nf, gen_1, pr_get_gen(pr));
    3223          21 :     res = nfpowmodideal(nf, res, stoi(g), cnd);
    3224             :   }
    3225             :   else { /* ell != p */
    3226         203 :     k = powis(stoi(ell),Z_lval(subiu(pr_norm(pr),1),ell));
    3227         203 :     k = divis(k,g);
    3228         203 :     modpr = nf_to_Fq_init(nf, &ppr, &T, &p);
    3229         203 :     (void)Fq_sqrtn(gen_1,k,T,p,&gen);
    3230         203 :     res = Fq_to_nf(gen, modpr);
    3231             :   }
    3232         224 :   return res;
    3233             : }
    3234             : 
    3235             : /* Ld[i] must be nontrivial powers of the same prime ell */
    3236             : /* pl : -1 at real places at which the extention must ramify, 0 elsewhere */
    3237             : GEN
    3238         168 : nfgwkummer(GEN nf, GEN Lpr, GEN Ld, GEN pl, long var)
    3239             : {
    3240         168 :   const long n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
    3241         168 :   pari_sp av = avma;
    3242             :   ulong ell;
    3243             :   long i, v;
    3244             :   GEN cnd, y, x, pol;
    3245         168 :   v = uisprimepower(n, &ell);
    3246         168 :   cnd = zeromatcopy(lg(Lpr)-1,2);
    3247             : 
    3248         168 :   y = vec_ei(lg(Lpr)-1,1);
    3249         413 :   for (i=1; i<lg(Lpr); i++) {
    3250         245 :     GEN pr = gel(Lpr,i), p = pr_get_p(pr), E;
    3251         245 :     long e = pr_get_e(pr);
    3252         245 :     gcoeff(cnd,i,1) = pr;
    3253             : 
    3254         245 :     if (!absequalui(ell,p))
    3255         217 :       E = gen_1;
    3256             :     else
    3257          28 :       E = addui(1 + v*e, divsi(e,subiu(p,1)));
    3258         245 :     gcoeff(cnd,i,2) = E;
    3259         245 :     gel(y,i) = localextdeg(nf, pr, idealpow(nf,pr,E), Ld[i], ell, n);
    3260             :   }
    3261             : 
    3262             :   /* TODO use a factoredextchinese to ease computations afterwards ? */
    3263         168 :   x = idealchinese(nf, mkvec2(cnd,pl), y);
    3264         168 :   x = basistoalg(nf,x);
    3265         168 :   pol = gsub(gpowgs(pol_x(var),n),x);
    3266             : 
    3267         168 :   return gerepileupto(av,pol);
    3268             : }
    3269             : 
    3270             : static GEN
    3271         707 : get_vecsmall(GEN v)
    3272             : {
    3273         707 :   switch(typ(v))
    3274             :   {
    3275         581 :     case t_VECSMALL: return v;
    3276         119 :     case t_VEC: if (RgV_is_ZV(v)) return ZV_to_zv(v);
    3277             :   }
    3278           7 :   pari_err_TYPE("nfgrunwaldwang",v);
    3279             :   return NULL;/*LCOV_EXCL_LINE*/
    3280             : }
    3281             : GEN
    3282         399 : nfgrunwaldwang(GEN nf0, GEN Lpr, GEN Ld, GEN pl, long var)
    3283             : {
    3284             :   ulong n, ell, ell2;
    3285         399 :   pari_sp av = avma;
    3286             :   GEN nf, bnf;
    3287             :   long t, w, i, vnf;
    3288             : 
    3289         399 :   if (var < 0) var = 0;
    3290         399 :   nf = get_nf(nf0,&t);
    3291         399 :   if (!nf) pari_err_TYPE("nfgrunwaldwang",nf0);
    3292         399 :   vnf = nf_get_varn(nf);
    3293         399 :   if (varncmp(var, vnf) >= 0)
    3294           7 :     pari_err_PRIORITY("nfgrunwaldwang", pol_x(var), ">=", vnf);
    3295         392 :   if (typ(Lpr) != t_VEC) pari_err_TYPE("nfgrunwaldwang",Lpr);
    3296         378 :   if (lg(Lpr) != lg(Ld)) pari_err_DIM("nfgrunwaldwang [#Lpr != #Ld]");
    3297         371 :   if (nf_get_degree(nf)==1) Lpr = shallowcopy(Lpr);
    3298         854 :   for (i=1; i<lg(Lpr); i++) {
    3299         490 :     GEN pr = gel(Lpr,i);
    3300         490 :     if (nf_get_degree(nf)==1 && typ(pr)==t_INT)
    3301          77 :       gel(Lpr,i) = gel(idealprimedec(nf,pr), 1);
    3302         413 :     else checkprid(pr);
    3303             :   }
    3304         364 :   if (lg(pl)-1 != nf_get_r1(nf))
    3305           7 :     pari_err_DOMAIN("nfgrunwaldwang [pl should have r1 components]", "#pl",
    3306           7 :         "!=", stoi(nf_get_r1(nf)), stoi(lg(pl)-1));
    3307             : 
    3308         357 :   Ld = get_vecsmall(Ld);
    3309         350 :   pl = get_vecsmall(pl);
    3310         350 :   bnf = get_bnf(nf0,&t);
    3311         350 :   n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
    3312             : 
    3313         350 :   if (!uisprimepower(n, &ell))
    3314           7 :     pari_err_IMPL("nfgrunwaldwang for non prime-power local degrees (a)");
    3315         791 :   for (i=1; i<lg(Ld); i++)
    3316         455 :     if (Ld[i]!=1 && (!uisprimepower(Ld[i],&ell2) || ell2!=ell))
    3317           7 :       pari_err_IMPL("nfgrunwaldwang for non prime-power local degrees (b)");
    3318         784 :   for (i=1; i<lg(pl); i++)
    3319         455 :     if (pl[i]==-1 && ell%2)
    3320           7 :       pari_err_IMPL("nfgrunwaldwang for non prime-power local degrees (c)");
    3321             : 
    3322         329 :   w = bnf? bnf_get_tuN(bnf): itos(gel(nfrootsof1(nf),1));
    3323             : 
    3324             :   /* TODO choice between kummer and generic ? Let user choose between speed
    3325             :    * and size */
    3326         329 :   if (w%n==0 && lg(Ld)>1)
    3327         168 :     return gerepileupto(av,nfgwkummer(nf,Lpr,Ld,pl,var));
    3328         161 :   if (ell==n) {
    3329         133 :     if (!bnf) bnf = Buchall(nf,0,0);
    3330         133 :     return gerepileupto(av,bnfgwgeneric(bnf,Lpr,Ld,pl,var));
    3331             :   }
    3332          28 :   pari_err_IMPL("nfgrunwaldwang for non-prime degree");
    3333             :   return NULL; /*LCOV_EXCL_LINE*/
    3334             : }
    3335             : 
    3336             : /** HASSE INVARIANTS **/
    3337             : 
    3338             : /* TODO long -> ulong + uel */
    3339             : static GEN
    3340         917 : hasseconvert(GEN H, long n)
    3341             : {
    3342             :   GEN h, c;
    3343             :   long i, l;
    3344         917 :   switch(typ(H)) {
    3345         847 :     case t_VEC:
    3346         847 :       l = lg(H); h = cgetg(l,t_VECSMALL);
    3347         847 :       if (l == 1) return h;
    3348         749 :       c = gel(H,1);
    3349         749 :       if (typ(c) == t_VEC && l == 3)
    3350         287 :         return mkvec2(gel(H,1),hasseconvert(gel(H,2),n));
    3351        1225 :       for (i=1; i<l; i++)
    3352             :       {
    3353         791 :         c = gel(H,i);
    3354         791 :         switch(typ(c)) {
    3355         567 :           case t_INT:  break;
    3356           7 :           case t_INTMOD:
    3357           7 :             c = gel(c,2); break;
    3358         196 :           case t_FRAC :
    3359         196 :             c = gmulgs(c,n);
    3360         196 :             if (typ(c) == t_INT) break;
    3361           7 :             pari_err_DOMAIN("hasseconvert [degree should be a denominator of the invariant]", "denom(h)", "ndiv", stoi(n), Q_denom(gel(H,i)));
    3362          21 :           default : pari_err_TYPE("Hasse invariant", c);
    3363             :         }
    3364         763 :         h[i] = smodis(c,n);
    3365             :       }
    3366         434 :       return h;
    3367          63 :     case t_VECSMALL: return H;
    3368             :   }
    3369           7 :   pari_err_TYPE("Hasse invariant", H);
    3370             :   return NULL;/*LCOV_EXCL_LINE*/
    3371             : }
    3372             : 
    3373             : /* assume f >= 2 */
    3374             : static long
    3375         392 : cyclicrelfrob0(GEN nf, GEN aut, GEN pr, GEN q, long f, long g)
    3376             : {
    3377         392 :   GEN T, p, a, b, modpr = nf_to_Fq_init(nf,&pr,&T,&p);
    3378             :   long s;
    3379             : 
    3380         392 :   a = pol_x(nf_get_varn(nf));
    3381         392 :   b = galoisapply(nf, aut, modpr_genFq(modpr));
    3382         392 :   b = nf_to_Fq(nf, b, modpr);
    3383        1253 :   for (s = 0; !ZX_equal(a, b); s++) a = Fq_pow(a, q, T, p);
    3384         392 :   return g * Fl_inv(s, f); /* < n */
    3385             : }
    3386             : 
    3387             : static long
    3388         938 : cyclicrelfrob(GEN rnf, GEN auts, GEN pr)
    3389             : {
    3390         938 :   pari_sp av = avma;
    3391         938 :   long f,g,frob, n = rnf_get_degree(rnf);
    3392         938 :   GEN P = rnfidealprimedec(rnf, pr);
    3393             : 
    3394         938 :   if (pr_get_e(gel(P,1)) > pr_get_e(pr))
    3395           0 :     pari_err_DOMAIN("cyclicrelfrob","e(PR/pr)",">",gen_1,pr);
    3396         938 :   g = lg(P) - 1;
    3397         938 :   f = n / g;
    3398             : 
    3399         938 :   if (f <= 2) frob = g % n;
    3400             :   else {
    3401         392 :     GEN nf2, PR = gel(P,1);
    3402         392 :     GEN autabs = rnfeltreltoabs(rnf,gel(auts,g));
    3403         392 :     nf2 = obj_check(rnf,rnf_NFABS);
    3404         392 :     autabs = nfadd(nf2, autabs, gmul(rnf_get_k(rnf), rnf_get_alpha(rnf)));
    3405         392 :     frob = cyclicrelfrob0(nf2, autabs, PR, pr_norm(pr), f, g);
    3406             :   }
    3407         938 :   return gc_long(av, frob);
    3408             : }
    3409             : 
    3410             : static long
    3411         553 : localhasse(GEN rnf, GEN cnd, GEN pl, GEN auts, GEN b, long k)
    3412             : {
    3413         553 :   pari_sp av = avma;
    3414             :   long v, m, h, lfa, frob, n, i;
    3415             :   GEN previous, y, pr, nf, q, fa;
    3416         553 :   nf = rnf_get_nf(rnf);
    3417         553 :   n = rnf_get_degree(rnf);
    3418         553 :   pr = gcoeff(cnd,k,1);
    3419         553 :   v = nfval(nf, b, pr);
    3420         553 :   m = lg(cnd)>1 ? nbrows(cnd) : 0;
    3421             : 
    3422             :   /* add the valuation of b to the conductor... */
    3423         553 :   previous = gcoeff(cnd,k,2);
    3424         553 :   gcoeff(cnd,k,2) = addis(previous, v);
    3425             : 
    3426         553 :   y = const_vec(m, gen_1);
    3427         553 :   gel(y,k) = b;
    3428             :   /* find a factored element y congruent to b mod pr^(vpr(b)+vpr(cnd)) and to 1 mod the conductor. */
    3429         553 :   y = factoredextchinese(nf, cnd, y, pl, &fa);
    3430         553 :   h = 0;
    3431         553 :   lfa = nbrows(fa);
    3432             :   /* sum of all Hasse invariants of (rnf/nf,aut,y) is 0, Hasse invariants at q!=pr are easy, Hasse invariant at pr is the same as for al=(rnf/nf,aut,b). */
    3433        1043 :   for (i=1; i<=lfa; i++) {
    3434         490 :     q = gcoeff(fa,i,1);
    3435         490 :     if (cmp_prime_ideal(pr,q)) {
    3436         455 :       frob = cyclicrelfrob(rnf, auts, q);
    3437         455 :       frob = Fl_mul(frob,umodiu(gcoeff(fa,i,2),n),n);
    3438         455 :       h = Fl_add(h,frob,n);
    3439             :     }
    3440             :   }
    3441             :   /* ...then restore it. */
    3442         553 :   gcoeff(cnd,k,2) = previous;
    3443         553 :   return gc_long(av, Fl_neg(h,n));
    3444             : }
    3445             : 
    3446             : static GEN
    3447         700 : allauts(GEN rnf, GEN aut)
    3448             : {
    3449         700 :   long n = rnf_get_degree(rnf), i;
    3450         700 :   GEN pol = rnf_get_pol(rnf), vaut;
    3451         700 :   if (n==1) n=2;
    3452         700 :   vaut = cgetg(n,t_VEC);
    3453         700 :   aut = lift_shallow(rnfbasistoalg(rnf,aut));
    3454         700 :   gel(vaut,1) = aut;
    3455        1008 :   for (i=1; i<n-1; i++)
    3456         308 :     gel(vaut,i+1) = RgX_rem(poleval(gel(vaut,i), aut), pol);
    3457         700 :   return vaut;
    3458             : }
    3459             : 
    3460             : static GEN
    3461         224 : clean_factor(GEN fa)
    3462             : {
    3463         224 :   GEN P2,E2, P = gel(fa,1), E = gel(fa,2);
    3464         224 :   long l = lg(P), i, j = 1;
    3465         224 :   P2 = cgetg(l, t_COL);
    3466         224 :   E2 = cgetg(l, t_COL);
    3467         609 :   for (i = 1;i < l; i++)
    3468         385 :     if (signe(gel(E,i))) {
    3469         252 :       gel(P2,j) = gel(P,i);
    3470         252 :       gel(E2,j) = gel(E,i); j++;
    3471             :     }
    3472         224 :   setlg(P2,j);
    3473         224 :   setlg(E2,j); return mkmat2(P2,E2);
    3474             : }
    3475             : 
    3476             : /* shallow concat x[1],...x[nx],y[1], ... y[ny], returning a t_COL. To be
    3477             :  * used when we do not know whether x,y are t_VEC or t_COL */
    3478             : static GEN
    3479         448 : colconcat(GEN x, GEN y)
    3480             : {
    3481         448 :   long i, lx = lg(x), ly = lg(y);
    3482         448 :   GEN z=cgetg(lx+ly-1, t_COL);
    3483         714 :   for (i=1; i<lx; i++) z[i]     = x[i];
    3484         952 :   for (i=1; i<ly; i++) z[lx+i-1]= y[i];
    3485         448 :   return z;
    3486             : }
    3487             : 
    3488             : /* return v(x) at all primes in listpr, replace x by cofactor */
    3489             : static GEN
    3490         924 : nfmakecoprime(GEN nf, GEN *px, GEN listpr)
    3491             : {
    3492         924 :   long j, l = lg(listpr);
    3493         924 :   GEN x1, x = *px, L = cgetg(l, t_COL);
    3494             : 
    3495         924 :   if (typ(x) != t_MAT)
    3496             :   { /* scalar, divide at the end (fast valuation) */
    3497         819 :     x1 = NULL;
    3498        1708 :     for (j=1; j<l; j++)
    3499             :     {
    3500         889 :       GEN pr = gel(listpr,j), e;
    3501         889 :       long v = nfval(nf, x, pr);
    3502         889 :       e = stoi(v); gel(L,j) = e;
    3503        1057 :       if (v) x1 = x1? idealmulpowprime(nf, x1, pr, e)
    3504         168 :                     : idealpow(nf, pr, e);
    3505             :     }
    3506         819 :     if (x1) x = idealdivexact(nf, idealhnf(nf,x), x1);
    3507             :   }
    3508             :   else
    3509             :   { /* HNF, divide as we proceed (reduce size) */
    3510         119 :     for (j=1; j<l; j++)
    3511             :     {
    3512          14 :       GEN pr = gel(listpr,j);
    3513          14 :       long v = idealval(nf, x, pr);
    3514          14 :       gel(L,j) = stoi(v);
    3515          14 :       if (v) x = idealmulpowprime(nf, x, pr, stoi(-v));
    3516             :     }
    3517             :   }
    3518         924 :   *px = x; return L;
    3519             : }
    3520             : 
    3521             : /* Caveat: factorizations are not sorted wrt cmp_prime_ideal: Lpr comes first */
    3522             : static GEN
    3523         224 : computecnd(GEN rnf, GEN Lpr)
    3524             : {
    3525             :   GEN id, nf, fa, Le, P,E;
    3526         224 :   long n = rnf_get_degree(rnf);
    3527             : 
    3528         224 :   nf = rnf_get_nf(rnf);
    3529         224 :   id = rnf_get_idealdisc(rnf);
    3530         224 :   Le = nfmakecoprime(nf, &id, Lpr);
    3531         224 :   fa = idealfactor(nf, id); /* part of D_{L/K} coprime with Lpr */
    3532         224 :   P =  colconcat(Lpr,gel(fa,1));
    3533         224 :   E =  colconcat(Le, gel(fa,2));
    3534         224 :   fa = mkmat2(P, gdiventgs(E, eulerphiu(n)));
    3535         224 :   return mkvec2(fa, clean_factor(fa));
    3536             : }
    3537             : 
    3538             : /* h >= 0 */
    3539             : static void
    3540           0 : nextgen(GEN gene, long h, GEN* gens, GEN* hgens, long* ngens, long* curgcd) {
    3541           0 :   long nextgcd = ugcd(h,*curgcd);
    3542           0 :   if (nextgcd == *curgcd) return;
    3543           0 :   (*ngens)++;
    3544           0 :   gel(*gens,*ngens) = gene;
    3545           0 :   gel(*hgens,*ngens) = utoi(h);
    3546           0 :   *curgcd = nextgcd;
    3547           0 :   return;
    3548             : }
    3549             : 
    3550             : static int
    3551           0 : dividesmod(long d, long h, long n) { return !(h%cgcd(d,n)); }
    3552             : 
    3553             : /* ramified prime with nontrivial Hasse invariant */
    3554             : static GEN
    3555           0 : localcomplete(GEN rnf, GEN pl, GEN cnd, GEN auts, long j, long n, long h, long* v)
    3556             : {
    3557             :   GEN nf, gens, hgens, pr, modpr, T, p, sol, U, D, b, gene, randg, pu;
    3558             :   long ngens, i, d, np, k, d1, d2, hg, dnf, vcnd, curgcd;
    3559           0 :   nf = rnf_get_nf(rnf);
    3560           0 :   pr = gcoeff(cnd,j,1);
    3561           0 :   np = umodiu(pr_norm(pr), n);
    3562           0 :   dnf = nf_get_degree(nf);
    3563           0 :   vcnd = itos(gcoeff(cnd,j,2));
    3564           0 :   ngens = 13+dnf;
    3565           0 :   gens = zerovec(ngens);
    3566           0 :   hgens = zerovec(ngens);
    3567           0 :   *v = 0;
    3568           0 :   curgcd = 0;
    3569           0 :   ngens = 0;
    3570             : 
    3571           0 :   if (!uisprime(n)) {
    3572           0 :     gene =  pr_get_gen(pr);
    3573           0 :     hg = localhasse(rnf, cnd, pl, auts, gene, j);
    3574           0 :     nextgen(gene, hg, &gens, &hgens, &ngens, &curgcd);
    3575             :   }
    3576             : 
    3577           0 :   if (ugcd(np,n) != 1) { /* GCD(Np,n) != 1 */
    3578           0 :     pu = idealprincipalunits(nf,pr,vcnd);
    3579           0 :     pu = abgrp_get_gen(pu);
    3580           0 :     for (i=1; i<lg(pu) && !dividesmod(curgcd,h,n); i++) {
    3581           0 :       gene = gel(pu,i);
    3582           0 :       hg = localhasse(rnf, cnd, pl, auts, gene, j);
    3583           0 :       nextgen(gene, hg, &gens, &hgens, &ngens, &curgcd);
    3584             :     }
    3585             :   }
    3586             : 
    3587           0 :   d = ugcd(np-1,n);
    3588           0 :   if (d != 1) { /* GCD(Np-1,n) != 1 */
    3589           0 :     modpr = nf_to_Fq_init(nf, &pr, &T, &p);
    3590           0 :     while (!dividesmod(curgcd,h,n)) { /* TODO gener_FpXQ_local */
    3591           0 :       if (T==NULL) randg = randomi(p);
    3592           0 :       else randg = random_FpX(degpol(T), varn(T),p);
    3593             : 
    3594           0 :       if (!gequal0(randg) && !gequal1(randg)) {
    3595           0 :         gene = Fq_to_nf(randg, modpr);
    3596           0 :         hg = localhasse(rnf, cnd, pl, auts, gene, j);
    3597           0 :         nextgen(gene, hg, &gens, &hgens, &ngens, &curgcd);
    3598             :       }
    3599             :     }
    3600             :   }
    3601             : 
    3602           0 :   setlg(gens,ngens+1);
    3603           0 :   setlg(hgens,ngens+1);
    3604             : 
    3605           0 :   sol = ZV_extgcd(hgens);
    3606           0 :   D = gel(sol,1);
    3607           0 :   U = gmael(sol,2,ngens);
    3608             : 
    3609           0 :   b = gen_1;
    3610           0 :   d = itou(D);
    3611           0 :   d1 = ugcd(d,n);
    3612           0 :   d2 = d/d1;
    3613           0 :   d = ((h/d1)*Fl_inv(d2,n))%n;
    3614           0 :   for (i=1; i<=ngens; i++) {
    3615           0 :     k = (itos(gel(U,i))*d)%n;
    3616           0 :     if (k<0) k = n-k;
    3617           0 :     if (k) b = nfmul(nf, b, nfpow_u(nf, gel(gens,i),k));
    3618           0 :     if (i==1) *v = k;
    3619             :   }
    3620           0 :   return b;
    3621             : }
    3622             : 
    3623             : static int
    3624         245 : testsplits(GEN data, GEN fa)
    3625             : {
    3626         245 :   GEN rnf = gel(data,1), forbid = gel(data,2), P = gel(fa,1), E = gel(fa,2);
    3627         245 :   long i, n, l = lg(P);
    3628             : 
    3629         336 :   for (i = 1; i < l; i++)
    3630             :   {
    3631         105 :     GEN pr = gel(P,i);
    3632         105 :     if (tablesearch(forbid, pr, &cmp_prime_ideal)) return 0;
    3633             :   }
    3634         231 :   n = rnf_get_degree(rnf);
    3635         308 :   for (i = 1; i < l; i++)
    3636             :   {
    3637          84 :     long e = itos(gel(E,i)) % n;
    3638          84 :     if (e)
    3639             :     {
    3640          77 :       GEN L = rnfidealprimedec(rnf, gel(P,i));
    3641          77 :       long g = lg(L) - 1;
    3642          77 :       if ((e * g) % n) return 0;
    3643             :     }
    3644             :   }
    3645         224 :   return 1;
    3646             : }
    3647             : 
    3648             : /* remove entries with Hasse invariant 0 */
    3649             : static GEN
    3650         476 : hassereduce(GEN hf)
    3651             : {
    3652         476 :   GEN pr,h, PR = gel(hf,1), H = gel(hf,2);
    3653         476 :   long i, j, l = lg(PR);
    3654             : 
    3655         476 :   pr= cgetg(l, t_VEC);
    3656         476 :   h = cgetg(l, t_VECSMALL);
    3657        1099 :   for (i = j = 1; i < l; i++)
    3658         623 :     if (H[i]) {
    3659         294 :       gel(pr,j) = gel(PR,i);
    3660         294 :       h[j] = H[i]; j++;
    3661             :     }
    3662         476 :   setlg(pr,j);
    3663         476 :   setlg(h,j); return mkvec2(pr,h);
    3664             : }
    3665             : 
    3666             : /* v vector of prid. Return underlying list of rational primes */
    3667             : static GEN
    3668         623 : pr_primes(GEN v)
    3669             : {
    3670         623 :   long i, l = lg(v);
    3671         623 :   GEN w = cgetg(l,t_VEC);
    3672        1729 :   for (i=1; i<l; i++) gel(w,i) = pr_get_p(gel(v,i));
    3673         623 :   return ZV_sort_uniq(w);
    3674             : }
    3675             : 
    3676             : /* rnf complete */
    3677             : static GEN
    3678         224 : alg_complete0(GEN rnf, GEN aut, GEN hf, GEN hi, long maxord)
    3679             : {
    3680         224 :   pari_sp av = avma;
    3681             :   GEN nf, pl, pl2, cnd, prcnd, cnds, y, Lpr, auts, b, fa, data, hfe;
    3682             :   GEN forbid, al, ind;
    3683             :   long D, n, d, i, j, l;
    3684         224 :   nf = rnf_get_nf(rnf);
    3685         224 :   n = rnf_get_degree(rnf);
    3686         224 :   d = nf_get_degree(nf);
    3687         224 :   D = d*n*n;
    3688         224 :   checkhasse(nf,hf,hi,n);
    3689         224 :   hf = hassereduce(hf);
    3690         224 :   Lpr = gel(hf,1);
    3691         224 :   hfe = gel(hf,2);
    3692             : 
    3693         224 :   auts = allauts(rnf,aut);
    3694             : 
    3695         224 :   pl = leafcopy(hi); /* conditions on the final b */
    3696         224 :   pl2 = leafcopy(hi); /* conditions for computing local Hasse invariants */
    3697         224 :   l = lg(pl); ind = cgetg(l, t_VECSMALL);
    3698         497 :   for (i = j = 1; i < l; i++)
    3699         273 :     if (hi[i]) { pl[i] = -1; pl2[i] = 1; } else ind[j++] = i;
    3700         224 :   setlg(ind, j);
    3701         224 :   y = nfpolsturm(nf, rnf_get_pol(rnf), ind);
    3702         420 :   for (i = 1; i < j; i++)
    3703         196 :     if (!signe(gel(y,i))) { pl[ind[i]] = 1; pl2[ind[i]] = 1; }
    3704             : 
    3705         224 :   cnds = computecnd(rnf,Lpr);
    3706         224 :   prcnd = gel(cnds,1);
    3707         224 :   cnd = gel(cnds,2);
    3708         224 :   y = cgetg(lgcols(prcnd),t_VEC);
    3709         224 :   forbid = vectrunc_init(lg(Lpr));
    3710         357 :   for (i=j=1; i<lg(Lpr); i++)
    3711             :   {
    3712         133 :     GEN pr = gcoeff(prcnd,i,1), yi;
    3713         133 :     long v, e = itou( gcoeff(prcnd,i,2) );
    3714         133 :     if (!e) {
    3715         133 :       long frob = cyclicrelfrob(rnf,auts,pr), f1 = ugcd(frob,n);
    3716         133 :       vectrunc_append(forbid, pr);
    3717         133 :       yi = gen_0;
    3718         133 :       v = ((hfe[i]/f1) * Fl_inv(frob/f1,n)) % n;
    3719             :     }
    3720             :     else
    3721           0 :       yi = localcomplete(rnf, pl2, cnd, auts, j++, n, hfe[i], &v);
    3722         133 :     gel(y,i) = yi;
    3723         133 :     gcoeff(prcnd,i,2) = stoi(e + v);
    3724             :   }
    3725         476 :   for (; i<lgcols(prcnd); i++) gel(y,i) = gen_1;
    3726         224 :   gen_sort_inplace(forbid, (void*)&cmp_prime_ideal, &cmp_nodata, NULL);
    3727         224 :   data = mkvec2(rnf,forbid);
    3728         224 :   b = factoredextchinesetest(nf,prcnd,y,pl,&fa,data,testsplits);
    3729             : 
    3730         224 :   al = cgetg(12, t_VEC);
    3731         224 :   gel(al,10)= gen_0; /* must be set first */
    3732         224 :   gel(al,1) = rnf;
    3733         224 :   gel(al,2) = auts;
    3734         224 :   gel(al,3) = basistoalg(nf,b);
    3735         224 :   gel(al,4) = hi;
    3736             :   /* add primes | disc or b with trivial Hasse invariant to hf */
    3737         224 :   Lpr = gel(prcnd,1); y = b;
    3738         224 :   (void)nfmakecoprime(nf, &y, Lpr);
    3739         224 :   Lpr = shallowconcat(Lpr, gel(idealfactor(nf,y), 1));
    3740         224 :   settyp(Lpr,t_VEC);
    3741         224 :   hf = mkvec2(Lpr, shallowconcat(hfe, const_vecsmall(lg(Lpr)-lg(hfe), 0)));
    3742         224 :   gel(al,5) = hf;
    3743         224 :   gel(al,6) = gen_0;
    3744         224 :   gel(al,7) = matid(D);
    3745         224 :   gel(al,8) = matid(D); /* TODO modify 7, 8 et 9 once LLL added */
    3746         224 :   gel(al,9) = algnatmultable(al,D);
    3747         224 :   gel(al,11)= algtracebasis(al);
    3748         224 :   if (maxord) al = alg_maximal_primes(al, pr_primes(Lpr));
    3749         224 :   return gerepilecopy(av, al);
    3750             : }
    3751             : 
    3752             : GEN
    3753           0 : alg_complete(GEN rnf, GEN aut, GEN hf, GEN hi, long maxord)
    3754             : {
    3755           0 :   long n = rnf_get_degree(rnf);
    3756           0 :   rnfcomplete(rnf);
    3757           0 :   return alg_complete0(rnf,aut,hasseconvert(hf,n),hasseconvert(hi,n), maxord);
    3758             : }
    3759             : 
    3760             : void
    3761        1239 : checkhasse(GEN nf, GEN hf, GEN hi, long n)
    3762             : {
    3763             :   GEN Lpr, Lh;
    3764             :   long i, sum;
    3765        1239 :   if (typ(hf) != t_VEC || lg(hf) != 3) pari_err_TYPE("checkhasse [hf]", hf);
    3766        1232 :   Lpr = gel(hf,1);
    3767        1232 :   Lh = gel(hf,2);
    3768        1232 :   if (typ(Lpr) != t_VEC) pari_err_TYPE("checkhasse [Lpr]", Lpr);
    3769        1232 :   if (typ(Lh) != t_VECSMALL) pari_err_TYPE("checkhasse [Lh]", Lh);
    3770        1232 :   if (typ(hi) != t_VECSMALL) pari_err_TYPE("checkhasse [hi]", hi);
    3771        1232 :   if ((nf && lg(hi) != nf_get_r1(nf)+1))
    3772           7 :     pari_err_DOMAIN("checkhasse [hi should have r1 components]","#hi","!=",stoi(nf_get_r1(nf)),stoi(lg(hi)-1));
    3773        1225 :   if (lg(Lpr) != lg(Lh))
    3774           7 :     pari_err_DIM("checkhasse [Lpr and Lh should have same length]");
    3775        2898 :   for (i=1; i<lg(Lpr); i++) checkprid(gel(Lpr,i));
    3776        1218 :   if (lg(gen_sort_uniq(Lpr, (void*)cmp_prime_ideal, cmp_nodata)) < lg(Lpr))
    3777           7 :     pari_err(e_MISC, "error in checkhasse [duplicate prime ideal]");
    3778        1211 :   sum = 0;
    3779        2877 :   for (i=1; i<lg(Lh); i++) sum = (sum+Lh[i])%n;
    3780        2611 :   for (i=1; i<lg(hi); i++) {
    3781        1414 :       if (hi[i] && 2*hi[i] != n) pari_err_DOMAIN("checkhasse", "Hasse invariant at real place [must be 0 or 1/2]", "!=", n%2? gen_0 : stoi(n/2), stoi(hi[i]));
    3782        1400 :       sum = (sum+hi[i])%n;
    3783             :   }
    3784        1197 :   if (sum<0) sum = n+sum;
    3785        1197 :   if (sum != 0)
    3786           7 :     pari_err_DOMAIN("checkhasse","sum(Hasse invariants)","!=",gen_0,Lh);
    3787        1190 : }
    3788             : 
    3789             : static GEN
    3790         322 : hassecoprime(GEN hf, GEN hi, long n)
    3791             : {
    3792         322 :   pari_sp av = avma;
    3793             :   long l, i, j, lk, inv;
    3794             :   GEN fa, P,E, res, hil, hfl;
    3795         322 :   hi = hasseconvert(hi, n);
    3796         308 :   hf = hasseconvert(hf, n);
    3797         287 :   checkhasse(NULL,hf,hi,n);
    3798         245 :   fa = factoru(n);
    3799         245 :   P = gel(fa,1); l = lg(P);
    3800         245 :   E = gel(fa,2);
    3801         245 :   res = cgetg(l,t_VEC);
    3802         497 :   for (i=1; i<l; i++) {
    3803         252 :     lk = upowuu(P[i],E[i]);
    3804         252 :     inv = Fl_invsafe((n/lk)%lk, lk);
    3805         252 :     hil = gcopy(hi);
    3806         252 :     hfl = gcopy(hf);
    3807             : 
    3808         252 :     if (P[i] == 2)
    3809         469 :       for (j=1; j<lg(hil); j++) hil[j] = hi[j]==0 ? 0 : lk/2;
    3810             :     else
    3811          91 :       for (j=1; j<lg(hil); j++) hil[j] = 0;
    3812         742 :     for (j=1; j<lgcols(hfl); j++) gel(hfl,2)[j] = (gel(hf,2)[j]*inv)%lk;
    3813         252 :     hfl = hassereduce(hfl);
    3814         252 :     gel(res,i) = mkvec3(hfl,hil,utoi(lk));
    3815             :   }
    3816             : 
    3817         245 :   return gerepilecopy(av, res);
    3818             : }
    3819             : 
    3820             : /* no garbage collection */
    3821             : static GEN
    3822          70 : genefrob(GEN nf, GEN gal, GEN r)
    3823             : {
    3824             :   long i;
    3825          70 :   GEN g = identity_perm(nf_get_degree(nf)), fa = Z_factor(r), p, pr, frob;
    3826         119 :   for (i=1; i<lgcols(fa); i++) {
    3827          49 :     p = gcoeff(fa,i,1);
    3828          49 :     pr = idealprimedec(nf, p);
    3829          49 :     pr = gel(pr,1);
    3830          49 :     frob = idealfrobenius(nf, gal, pr);
    3831          49 :     g = perm_mul(g, perm_pow(frob, itos(gcoeff(fa,i,2))));
    3832             :   }
    3833          70 :   return g;
    3834             : }
    3835             : 
    3836             : static GEN
    3837         224 : rnfcycaut(GEN rnf)
    3838             : {
    3839         224 :   GEN nf2 = obj_check(rnf, rnf_NFABS);
    3840             :   GEN L, alpha, pol, salpha, s, sj, polabs, k, X, pol0, nf;
    3841             :   long i, d, j;
    3842         224 :   d = rnf_get_degree(rnf);
    3843         224 :   L = galoisconj(nf2,NULL);
    3844         224 :   alpha = lift_shallow(rnf_get_alpha(rnf));
    3845         224 :   pol = rnf_get_pol(rnf);
    3846         224 :   k = rnf_get_k(rnf);
    3847         224 :   polabs = rnf_get_polabs(rnf);
    3848         224 :   nf = rnf_get_nf(rnf);
    3849         224 :   pol0 = nf_get_pol(nf);
    3850         224 :   X = RgX_rem(pol_x(varn(pol0)), pol0);
    3851             : 
    3852             :   /* TODO check mod prime of degree 1 */
    3853         301 :   for (i=1; i<lg(L); i++) {
    3854         301 :     s = gel(L,i);
    3855         301 :     salpha = RgX_RgXQ_eval(alpha,s,polabs);
    3856         301 :     if (!gequal(alpha,salpha)) continue;
    3857             : 
    3858         273 :     s = lift_shallow(rnfeltabstorel(rnf,s));
    3859         273 :     sj = s = gsub(s, gmul(k,X));
    3860         532 :     for (j=1; !gequal0(gsub(sj,pol_x(varn(s)))); j++)
    3861         259 :       sj = RgX_RgXQ_eval(sj,s,pol);
    3862         273 :     if (j<d) continue;
    3863         224 :     return s;
    3864             :   }
    3865             :   return NULL; /*LCOV_EXCL_LINE*/
    3866             : }
    3867             : 
    3868             : /* returns Lpr augmented with an extra, distinct prime */
    3869             : /* TODO be less lazy and return a small prime */
    3870             : static GEN
    3871          84 : extraprime(GEN nf, GEN Lpr)
    3872             : {
    3873          84 :   GEN Lpr2, p = gen_2, pr;
    3874             :   long i;
    3875          84 :   Lpr2 = cgetg(lg(Lpr)+1,t_VEC);
    3876          98 :   for (i=1; i<lg(Lpr); i++)
    3877             :   {
    3878          14 :     gel(Lpr2,i) = gel(Lpr,i);
    3879          14 :     p = gmax_shallow(p, pr_get_p(gel(Lpr,i)));
    3880             :   }
    3881          84 :   p = nextprime(addis(p,1));
    3882          84 :   pr = gel(idealprimedec_limit_f(nf, p, 0), 1);
    3883          84 :   gel(Lpr2,lg(Lpr)) = pr;
    3884          84 :   return Lpr2;
    3885             : }
    3886             : 
    3887             : GEN
    3888         336 : alg_hasse(GEN nf, long n, GEN hf, GEN hi, long var, long maxord)
    3889             : {
    3890         336 :   pari_sp av = avma;
    3891         336 :   GEN primary, al = gen_0, al2, rnf, hil, hfl, Ld, pl, pol, Lpr, aut, Lpr2, Ld2;
    3892             :   long i, lk, j, maxdeg;
    3893         336 :   dbg_printf(1)("alg_hasse\n");
    3894         336 :   if (n<=1) pari_err_DOMAIN("alg_hasse", "degree", "<=", gen_1, stoi(n));
    3895         322 :   primary = hassecoprime(hf, hi, n);
    3896         476 :   for (i=1; i<lg(primary); i++) {
    3897         252 :     lk = itos(gmael(primary,i,3));
    3898         252 :     hfl = gmael(primary,i,1);
    3899         252 :     hil = gmael(primary,i,2);
    3900         252 :     checkhasse(nf, hfl, hil, lk);
    3901         245 :     dbg_printf(1)("alg_hasse: i=%d hf=%Ps hi=%Ps lk=%d\n", i, hfl, hil, lk);
    3902             : 
    3903         245 :     if (lg(gel(hfl,1))>1 || lk%2==0) {
    3904         238 :       maxdeg = 1;
    3905         238 :       Lpr = gel(hfl,1);
    3906         238 :       Ld = gcopy(gel(hfl,2));
    3907         385 :       for (j=1; j<lg(Ld); j++)
    3908             :       {
    3909         147 :         Ld[j] = lk/ugcd(lk,Ld[j]);
    3910         147 :         maxdeg = maxss(Ld[j],maxdeg);
    3911             :       }
    3912         238 :       pl = gcopy(hil);
    3913         525 :       for (j=1; j<lg(pl); j++) if(pl[j])
    3914             :       {
    3915          77 :         pl[j] = -1;
    3916          77 :         maxdeg = maxss(maxdeg,2);
    3917             :       }
    3918             : 
    3919         238 :       Lpr2 = Lpr;
    3920         238 :       Ld2 = Ld;
    3921         238 :       if (maxdeg<lk)
    3922             :       {
    3923         154 :         if (maxdeg==1 && lk==2 && lg(pl)>1) pl[1] = -1;
    3924             :         else
    3925             :         {
    3926          84 :           Lpr2 = extraprime(nf,Lpr);
    3927          84 :           Ld2 = cgetg(lg(Ld)+1, t_VECSMALL);
    3928          98 :           for (j=1; j<lg(Ld); j++) Ld2[j] = Ld[j];
    3929          84 :           Ld2[lg(Ld)] = lk;
    3930             :         }
    3931             :       }
    3932             : 
    3933         238 :       dbg_printf(2)("alg_hasse: calling nfgrunwaldwang Lpr=%Ps Pd=%Ps pl=%Ps\n",
    3934             :           Lpr, Ld, pl);
    3935         238 :       pol = nfgrunwaldwang(nf, Lpr2, Ld2, pl, var);
    3936         224 :       dbg_printf(2)("alg_hasse: calling rnfinit(%Ps)\n", pol);
    3937         224 :       rnf = rnfinit0(nf,pol,1);
    3938         224 :       dbg_printf(2)("alg_hasse: computing automorphism\n");
    3939         224 :       aut = rnfcycaut(rnf);
    3940         224 :       dbg_printf(2)("alg_hasse: calling alg_complete\n");
    3941         224 :       al2 = alg_complete0(rnf,aut,hfl,hil,maxord);
    3942             :     }
    3943           7 :     else al2 = alg_matrix(nf, lk, var, cgetg(1,t_VEC), maxord);
    3944             : 
    3945         231 :     if (i==1) al = al2;
    3946           7 :     else      al = algtensor(al,al2,maxord);
    3947             :   }
    3948         224 :   return gerepilecopy(av,al);
    3949             : }
    3950             : 
    3951             : /** CYCLIC ALGEBRA WITH GIVEN HASSE INVARIANTS **/
    3952             : 
    3953             : /* no garbage collection */
    3954             : static int
    3955          70 : linindep(GEN pol, GEN L)
    3956             : {
    3957             :   long i;
    3958             :   GEN fa;
    3959          70 :   for (i=1; i<lg(L); i++) {
    3960           0 :     fa = nffactor(gel(L,i),pol);
    3961           0 :     if (lgcols(fa)>2) return 0;
    3962             :   }
    3963          70 :   return 1;
    3964             : }
    3965             : 
    3966             : /* no garbage collection */
    3967             : static GEN
    3968          70 : subcycloindep(GEN nf, long n, long v, GEN L, GEN *pr)
    3969             : {
    3970             :   pari_sp av;
    3971             :   forprime_t S;
    3972             :   ulong p;
    3973          70 :   u_forprime_arith_init(&S, 1, ULONG_MAX, 1, n);
    3974          70 :   av = avma;
    3975          77 :   while ((p = u_forprime_next(&S)))
    3976             :   {
    3977          77 :     ulong r = pgener_Fl(p);
    3978          77 :     GEN pol = galoissubcyclo(utoipos(p), utoipos(Fl_powu(r,n,p)), 0, v);
    3979          77 :     GEN fa = nffactor(nf, pol);
    3980          77 :     if (lgcols(fa) == 2 && linindep(pol,L)) { *pr = utoipos(r); return pol; }
    3981           7 :     set_avma(av);
    3982             :   }
    3983             :   pari_err_BUG("subcycloindep (no suitable prime = 1(mod n))"); /*LCOV_EXCL_LINE*/
    3984             :   *pr = NULL; return NULL; /*LCOV_EXCL_LINE*/
    3985             : }
    3986             : 
    3987             : GEN
    3988          77 : alg_matrix(GEN nf, long n, long v, GEN L, long maxord)
    3989             : {
    3990          77 :   pari_sp av = avma;
    3991             :   GEN pol, gal, rnf, cyclo, g, r, aut;
    3992          77 :   dbg_printf(1)("alg_matrix\n");
    3993          77 :   if (n<=0) pari_err_DOMAIN("alg_matrix", "n", "<=", gen_0, stoi(n));
    3994          70 :   pol = subcycloindep(nf, n, v, L, &r);
    3995          70 :   rnf = rnfinit(nf, pol);
    3996          70 :   cyclo = nfinit(pol, nf_get_prec(nf));
    3997          70 :   gal = galoisinit(cyclo, NULL);
    3998          70 :   g = genefrob(cyclo,gal,r);
    3999          70 :   aut = galoispermtopol(gal,g);
    4000          70 :   return gerepileupto(av, alg_cyclic(rnf, aut, gen_1, maxord));
    4001             : }
    4002             : 
    4003             : GEN
    4004         273 : alg_hilbert(GEN nf, GEN a, GEN b, long v, long maxord)
    4005             : {
    4006         273 :   pari_sp av = avma;
    4007             :   GEN C, P, rnf, aut;
    4008         273 :   dbg_printf(1)("alg_hilbert\n");
    4009         273 :   checknf(nf);
    4010         273 :   if (!isint1(Q_denom(a)))
    4011           7 :     pari_err_DOMAIN("alg_hilbert", "denominator(a)", "!=", gen_1,a);
    4012         266 :   if (!isint1(Q_denom(b)))
    4013           7 :     pari_err_DOMAIN("alg_hilbert", "denominator(b)", "!=", gen_1,b);
    4014             : 
    4015         259 :   if (v < 0) v = 0;
    4016         259 :   C = Rg_col_ei(gneg(a), 3, 3);
    4017         259 :   gel(C,1) = gen_1;
    4018         259 :   P = gtopoly(C,v);
    4019         259 :   rnf = rnfinit(nf, P);
    4020         252 :   aut = gneg(pol_x(v));
    4021         252 :   return gerepileupto(av, alg_cyclic(rnf, aut, b, maxord));
    4022             : }
    4023             : 
    4024             : GEN
    4025        1043 : alginit(GEN A, GEN B, long v, long maxord)
    4026             : {
    4027             :   long w;
    4028        1043 :   switch(nftyp(A))
    4029             :   {
    4030         875 :     case typ_NF:
    4031         875 :       if (v<0) v=0;
    4032         875 :       w = gvar(nf_get_pol(A));
    4033         875 :       if (varncmp(v,w)>=0) pari_err_PRIORITY("alginit", pol_x(v), ">=", w);
    4034         861 :       switch(typ(B))
    4035             :       {
    4036             :         long nB;
    4037          70 :         case t_INT: return alg_matrix(A, itos(B), v, cgetg(1,t_VEC), maxord);
    4038         784 :         case t_VEC:
    4039         784 :           nB = lg(B)-1;
    4040         784 :           if (nB && typ(gel(B,1)) == t_MAT) return alg_csa_table(A,B,v,maxord);
    4041             :           switch(nB)
    4042             :           {
    4043         273 :             case 2: return alg_hilbert(A, gel(B,1), gel(B,2), v, maxord);
    4044         343 :             case 3:
    4045         343 :               if (typ(gel(B,1))!=t_INT)
    4046           7 :                   pari_err_TYPE("alginit [degree should be an integer]", gel(B,1));
    4047         336 :               return alg_hasse(A, itos(gel(B,1)), gel(B,2), gel(B,3), v,
    4048             :                                                                       maxord);
    4049             :           }
    4050             :       }
    4051          14 :       pari_err_TYPE("alginit", B); break;
    4052             : 
    4053         161 :     case typ_RNF:
    4054         161 :       if (typ(B) != t_VEC || lg(B) != 3) pari_err_TYPE("alginit", B);
    4055         147 :       return alg_cyclic(A, gel(B,1), gel(B,2), maxord);
    4056             :   }
    4057           7 :   pari_err_TYPE("alginit", A);
    4058             :   return NULL;/*LCOV_EXCL_LINE*/
    4059             : }
    4060             : 
    4061             : /* assumes al CSA or CYCLIC */
    4062             : static GEN
    4063         833 : algnatmultable(GEN al, long D)
    4064             : {
    4065             :   GEN res, x;
    4066             :   long i;
    4067         833 :   res = cgetg(D+1,t_VEC);
    4068        9793 :   for (i=1; i<=D; i++) {
    4069        8960 :     x = algnattoalg(al,col_ei(D,i));
    4070        8960 :     gel(res,i) = algZmultable(al,x);
    4071             :   }
    4072         833 :   return res;
    4073             : }
    4074             : 
    4075             : /* no garbage collection */
    4076             : static void
    4077         476 : algcomputehasse(GEN al)
    4078             : {
    4079             :   long r1, k, n, m, m1, m2, m3, i, m23, m123;
    4080             :   GEN rnf, nf, b, fab, disc2, cnd, fad, auts, pr, pl, perm, y, hi, PH, H, L;
    4081             : 
    4082         476 :   rnf = alg_get_splittingfield(al);
    4083         476 :   n = rnf_get_degree(rnf);
    4084         476 :   nf = rnf_get_nf(rnf);
    4085         476 :   b = alg_get_b(al);
    4086         476 :   r1 = nf_get_r1(nf);
    4087         476 :   auts = alg_get_auts(al);
    4088         476 :   (void)alg_get_abssplitting(al);
    4089             : 
    4090         476 :   y = nfpolsturm(nf, rnf_get_pol(rnf), NULL);
    4091         476 :   pl = cgetg(r1+1, t_VECSMALL);
    4092             :   /* real places where rnf/nf ramifies */
    4093        1001 :   for (k = 1; k <= r1; k++) pl[k] = !signe(gel(y,k));
    4094             : 
    4095             :   /* infinite Hasse invariants */
    4096         476 :   if (odd(n)) hi = const_vecsmall(r1, 0);
    4097             :   else
    4098             :   {
    4099         406 :     GEN s = nfsign(nf, b);
    4100         406 :     hi = cgetg(r1+1, t_VECSMALL);
    4101         882 :     for (k = 1; k<=r1; k++) hi[k] = (s[k] && pl[k]) ? (n/2) : 0;
    4102             :   }
    4103             : 
    4104         476 :   fab = idealfactor(nf, b);
    4105         476 :   disc2 = rnf_get_idealdisc(rnf);
    4106         476 :   L = nfmakecoprime(nf, &disc2, gel(fab,1));
    4107         476 :   m = lg(L)-1;
    4108             :   /* m1 = #{pr|b: pr \nmid disc}, m3 = #{pr|b: pr | disc} */
    4109         476 :   perm = cgetg(m+1, t_VECSMALL);
    4110         861 :   for (i=1, m1=m, k=1; k<=m; k++)
    4111         385 :     if (signe(gel(L,k))) perm[m1--] = k; else perm[i++] = k;
    4112         476 :   m3 = m - m1;
    4113             : 
    4114             :   /* disc2 : factor of disc coprime to b */
    4115         476 :   fad = idealfactor(nf, disc2);
    4116             :   /* m2 : number of prime factors of disc not dividing b */
    4117         476 :   m2 = nbrows(fad);
    4118         476 :   m23 = m2+m3;
    4119         476 :   m123 = m1+m2+m3;
    4120             : 
    4121             :   /* initialize the possibly ramified primes (hasse) and the factored conductor of rnf/nf (cnd) */
    4122         476 :   cnd = zeromatcopy(m23,2);
    4123         476 :   PH = cgetg(m123+1, t_VEC); /* ramified primes */
    4124         476 :   H = cgetg(m123+1, t_VECSMALL); /* Hasse invariant */
    4125             :   /* compute Hasse invariant at primes that are unramified in rnf/nf */
    4126         826 :   for (k=1; k<=m1; k++) {/* pr | b, pr \nmid disc */
    4127         350 :     long frob, e, j = perm[k];
    4128         350 :     pr = gcoeff(fab,j,1);
    4129         350 :     e = itos(gcoeff(fab,j,2));
    4130         350 :     frob = cyclicrelfrob(rnf, auts, pr);
    4131         350 :     gel(PH,k) = pr;
    4132         350 :     H[k] = Fl_mul(frob, e, n);
    4133             :   }
    4134             :   /* compute Hasse invariant at primes that are ramified in rnf/nf */
    4135         994 :   for (k=1; k<=m2; k++) {/* pr \nmid b, pr | disc */
    4136         518 :     pr = gcoeff(fad,k,1);
    4137         518 :     gel(PH,k+m1) = pr;
    4138         518 :     gcoeff(cnd,k,1) = pr;
    4139         518 :     gcoeff(cnd,k,2) = gcoeff(fad,k,2);
    4140             :   }
    4141         511 :   for (k=1; k<=m3; k++) { /* pr | (b, disc) */
    4142          35 :     long j = perm[k+m1];
    4143          35 :     pr = gcoeff(fab,j,1);
    4144          35 :     gel(PH,k+m1+m2) = pr;
    4145          35 :     gcoeff(cnd,k+m2,1) = pr;
    4146          35 :     gcoeff(cnd,k+m2,2) = gel(L,j);
    4147             :   }
    4148         476 :   gel(cnd,2) = gdiventgs(gel(cnd,2), eulerphiu(n));
    4149        1029 :   for (k=1; k<=m23; k++) H[k+m1] = localhasse(rnf, cnd, pl, auts, b, k);
    4150         476 :   gel(al,4) = hi;
    4151         476 :   perm = gen_indexsort(PH, (void*)&cmp_prime_ideal, &cmp_nodata);
    4152         476 :   gel(al,5) = mkvec2(vecpermute(PH,perm),vecsmallpermute(H,perm));
    4153         476 :   checkhasse(nf,alg_get_hasse_f(al),alg_get_hasse_i(al),n);
    4154         476 : }
    4155             : 
    4156             : static GEN
    4157         749 : alg_maximal_primes(GEN al, GEN P)
    4158             : {
    4159         749 :   pari_sp av = avma;
    4160         749 :   long l = lg(P), i;
    4161        1939 :   for (i=1; i<l; i++)
    4162             :   {
    4163        1190 :     if (i != 1) al = gerepilecopy(av, al);
    4164        1190 :     al = alg_pmaximal(al,gel(P,i));
    4165             :   }
    4166         749 :   return al;
    4167             : }
    4168             : 
    4169             : GEN
    4170         483 : alg_cyclic(GEN rnf, GEN aut, GEN b, long maxord)
    4171             : {
    4172         483 :   pari_sp av = avma;
    4173             :   GEN al, nf;
    4174             :   long D, n, d;
    4175         483 :   dbg_printf(1)("alg_cyclic\n");
    4176         483 :   checkrnf(rnf);
    4177         483 :   if (!isint1(Q_denom(b)))
    4178           7 :     pari_err_DOMAIN("alg_cyclic", "denominator(b)", "!=", gen_1,b);
    4179             : 
    4180         476 :   nf = rnf_get_nf(rnf);
    4181         476 :   n = rnf_get_degree(rnf);
    4182         476 :   d = nf_get_degree(nf);
    4183         476 :   D = d*n*n;
    4184             : 
    4185         476 :   al = cgetg(12,t_VEC);
    4186         476 :   gel(al,10)= gen_0; /* must be set first */
    4187         476 :   gel(al,1) = rnf;
    4188         476 :   gel(al,2) = allauts(rnf, aut);
    4189         476 :   gel(al,3) = basistoalg(nf,b);
    4190         476 :   rnf_build_nfabs(rnf, nf_get_prec(nf));
    4191         476 :   gel(al,6) = gen_0;
    4192         476 :   gel(al,7) = matid(D);
    4193         476 :   gel(al,8) = matid(D); /* TODO modify 7, 8 et 9 once LLL added */
    4194         476 :   gel(al,9) = algnatmultable(al,D);
    4195         476 :   gel(al,11)= algtracebasis(al);
    4196             : 
    4197         476 :   algcomputehasse(al);
    4198             : 
    4199         476 :   if (maxord) {
    4200         413 :     GEN hf = alg_get_hasse_f(al), pr = gel(hf,1);
    4201         413 :     al = alg_maximal_primes(al, pr_primes(pr));
    4202             :   }
    4203         476 :   return gerepilecopy(av, al);
    4204             : }
    4205             : 
    4206             : static int
    4207         378 : ismaximalsubfield(GEN al, GEN x, GEN d, long v, GEN *pt_minpol)
    4208             : {
    4209         378 :   GEN cp = algbasischarpoly(al, x, v), lead;
    4210         378 :   if (!ispower(cp, d, pt_minpol)) return 0;
    4211         378 :   lead = leading_coeff(*pt_minpol);
    4212         378 :   if (isintm1(lead)) *pt_minpol = gneg(*pt_minpol);
    4213         378 :   return ZX_is_irred(*pt_minpol);
    4214             : }
    4215             : 
    4216             : static GEN
    4217         133 : findmaximalsubfield(GEN al, GEN d, long v)
    4218             : {
    4219         133 :   long count, nb=2, i, N = alg_get_absdim(al), n = nf_get_degree(alg_get_center(al));
    4220         133 :   GEN x, minpol, maxc = gen_1;
    4221             : 
    4222         210 :   for (i=n+1; i<=N; i+=n) {
    4223         336 :     for (count=0; count<2 && i+count<=N; count++) {
    4224         259 :       x = col_ei(N,i+count);
    4225         259 :       if (ismaximalsubfield(al, x, d, v, &minpol)) return mkvec2(x,minpol);
    4226             :     }
    4227             :   }
    4228             : 
    4229             :   while(1) {
    4230         119 :     x = zerocol(N);
    4231         504 :     for (count=0; count<nb; count++)
    4232             :     {
    4233         385 :       i = random_Fl(N)+1;
    4234         385 :       gel(x,i) = addiu(randomi(maxc),1);
    4235         385 :       if (random_bits(1)) gel(x,i) = negi(gel(x,i));
    4236             :     }
    4237         119 :     if (ismaximalsubfield(al, x, d, v, &minpol)) return mkvec2(x,minpol);
    4238          56 :     if (!random_bits(3)) maxc = addiu(maxc,1);
    4239          56 :     if (nb<N) nb++;
    4240             :   }
    4241             : 
    4242             :   return NULL; /* LCOV_EXCL_LINE */
    4243             : }
    4244             : 
    4245             : static GEN
    4246         133 : frobeniusform(GEN al, GEN x)
    4247             : {
    4248             :   GEN M, FP, P, Pi;
    4249             : 
    4250             :   /* /!\ has to be the *right* multiplication table */
    4251         133 :   M = algbasisrightmultable(al, x);
    4252             : 
    4253         133 :   FP = matfrobenius(M,2,0); /* M = P^(-1)*F*P */
    4254         133 :   P = gel(FP,2);
    4255         133 :   Pi = RgM_inv(P);
    4256         133 :   return mkvec2(P, Pi);
    4257             : }
    4258             : 
    4259             : static void
    4260         133 : computesplitting(GEN al, long d, long v)
    4261             : {
    4262         133 :   GEN subf, x, pol, polabs, basis, P, Pi, nf = alg_get_center(al), rnf, Lbasis, Lbasisinv, Q, pows;
    4263         133 :   long i, n = nf_get_degree(nf), nd = n*d, N = alg_get_absdim(al), j, j2;
    4264             : 
    4265         133 :   subf = findmaximalsubfield(al, utoipos(d), v);
    4266         133 :   x = gel(subf, 1);
    4267         133 :   polabs = gel(subf, 2);
    4268             : 
    4269             :   /* Frobenius form to obtain L-vector space structure */
    4270         133 :   basis = frobeniusform(al, x);
    4271         133 :   P = gel(basis, 1);
    4272         133 :   Pi = gel(basis, 2);
    4273             : 
    4274             :   /* construct rnf of splitting field */
    4275         133 :   pol = nffactor(nf,polabs);
    4276         133 :   pol = gcoeff(pol,1,1);
    4277         133 :   gel(al,1) = rnf = rnfinit(nf, pol);
    4278             :   /* since pol is irreducible over Q, we have k=0 in rnf. */
    4279         133 :   if (!gequal0(rnf_get_k(rnf)))
    4280             :     pari_err_BUG("computesplitting (k!=0)"); /*LCOV_EXCL_LINE*/
    4281         133 :   gel(al,6) = gen_0;
    4282         133 :   rnf_build_nfabs(rnf, nf_get_prec(nf));
    4283             : 
    4284             :   /* construct splitting data */
    4285         133 :   Lbasis = cgetg(d+1, t_MAT);
    4286         357 :   for (j=j2=1; j<=d; j++, j2+=nd)
    4287         224 :     gel(Lbasis,j) = gel(Pi,j2);
    4288             : 
    4289         133 :   Q = zeromatcopy(d,N);
    4290         133 :   pows = pol_x_powers(nd,v);
    4291         357 :   for (i=j=1; j<=N; j+=nd, i++)
    4292        1085 :   for (j2=0; j2<nd; j2++)
    4293         861 :     gcoeff(Q,i,j+j2) = mkpolmod(gel(pows,j2+1),polabs);
    4294         133 :   Lbasisinv = RgM_mul(Q,P);
    4295             : 
    4296         133 :   gel(al,3) = mkvec3(x,Lbasis,Lbasisinv);
    4297         133 : }
    4298             : 
    4299             : /* assumes that mt defines a central simple algebra over nf */
    4300             : GEN
    4301         161 : alg_csa_table(GEN nf, GEN mt0, long v, long maxord)
    4302             : {
    4303         161 :   pari_sp av = avma;
    4304             :   GEN al, mt;
    4305         161 :   long n, D, d2 = lg(mt0)-1, d = usqrt(d2);
    4306         161 :   dbg_printf(1)("alg_csa_table\n");
    4307             : 
    4308         161 :   nf = checknf(nf);
    4309         161 :   mt = check_relmt(nf,mt0);
    4310         147 :   if (!mt) pari_err_TYPE("alg_csa_table", mt0);
    4311         140 :   n = nf_get_degree(nf);
    4312         140 :   D = n*d2;
    4313         140 :   if (d*d != d2)
    4314           7 :     pari_err_DOMAIN("alg_csa_table","(nonsquare) dimension","!=",stoi(d*d),mt);
    4315             : 
    4316         133 :   al = cgetg(12, t_VEC);
    4317         133 :   gel(al,10) = gen_0; /* must be set first */
    4318         133 :   gel(al,1) = zerovec(12); gmael(al,1,10) = nf;
    4319         133 :   gmael(al,1,1) = gpowgs(pol_x(0), d); /* placeholder before splitting field */
    4320         133 :   gel(al,2) = mt;
    4321         133 :   gel(al,3) = gen_0; /* placeholder */
    4322         133 :   gel(al,4) = gel(al,5) = gen_0; /* TODO Hasse invariants */
    4323         133 :   gel(al,5) = gel(al,6) = gen_0; /* placeholder */
    4324         133 :   gel(al,7) = matid(D);
    4325         133 :   gel(al,8) = matid(D);
    4326         133 :   gel(al,9) = algnatmultable(al,D);
    4327         133 :   gel(al,11)= algtracebasis(al);
    4328         133 :   if (maxord) al = alg_maximal(al);
    4329         133 :   computesplitting(al, d, v);
    4330         133 :   return gerepilecopy(av, al);
    4331             : }
    4332             : 
    4333             : static GEN
    4334       36337 : algtableinit_i(GEN mt0, GEN p)
    4335             : {
    4336             :   GEN al, mt;
    4337             :   long i, n;
    4338             : 
    4339       36337 :   if (p && !signe(p)) p = NULL;
    4340       36337 :   mt = check_mt(mt0,p);
    4341       36337 :   if (!mt) pari_err_TYPE("algtableinit", mt0);
    4342       36337 :   if (!p && !isint1(Q_denom(mt0)))
    4343           7 :     pari_err_DOMAIN("algtableinit", "denominator(mt)", "!=", gen_1, mt0);
    4344       36330 :   n = lg(mt)-1;
    4345       36330 :   al = cgetg(12, t_VEC);
    4346      254310 :   for (i=1; i<=6; i++) gel(al,i) = gen_0;
    4347       36330 :   gel(al,7) = matid(n);
    4348       36330 :   gel(al,8) = matid(n);
    4349       36330 :   gel(al,9) = mt;
    4350       36330 :   gel(al,10) = p? p: gen_0;
    4351       36330 :   gel(al,11)= algtracebasis(al);
    4352       36330 :   return al;
    4353             : }
    4354             : GEN
    4355        4193 : algtableinit(GEN mt0, GEN p)
    4356             : {
    4357        4193 :   pari_sp av = avma;
    4358        4193 :   if (p)
    4359             :   {
    4360        4074 :     if (typ(p) != t_INT) pari_err_TYPE("algtableinit",p);
    4361        4067 :     if (signe(p) && !BPSW_psp(p)) pari_err_PRIME("algtableinit",p);
    4362             :   }
    4363        4172 :   return gerepilecopy(av, algtableinit_i(mt0, p));
    4364             : }
    4365             : 
    4366             : /** REPRESENTATIONS OF GROUPS **/
    4367             : 
    4368             : static GEN
    4369         294 : list_to_regular_rep(GEN elts, long n)
    4370             : {
    4371             :   GEN reg, elts2, g;
    4372             :   long i,j;
    4373         294 :   elts = shallowcopy(elts);
    4374         294 :   gen_sort_inplace(elts, (void*)&vecsmall_lexcmp, &cmp_nodata, NULL);
    4375         294 :   reg = cgetg(n+1, t_VEC);
    4376         294 :   gel(reg,1) = identity_perm(n);
    4377        3857 :   for (i=2; i<=n; i++) {
    4378        3563 :     g = perm_inv(gel(elts,i));
    4379        3563 :     elts2 = cgetg(n+1, t_VEC);
    4380       74543 :     for (j=1; j<=n; j++) gel(elts2,j) = perm_mul(g,gel(elts,j));
    4381        3563 :     gen_sort_inplace(elts2, (void*)&vecsmall_lexcmp, &cmp_nodata, &gel(reg,i));
    4382             :   }
    4383         294 :   return reg;
    4384             : }
    4385             : 
    4386             : static GEN
    4387        3857 : matrix_perm(GEN perm, long n)
    4388             : {
    4389             :   GEN m;
    4390             :   long j;
    4391        3857 :   m = cgetg(n+1, t_MAT);
    4392       78694 :   for (j=1; j<=n; j++) {
    4393       74837 :     gel(m,j) = col_ei(n,perm[j]);
    4394             :   }
    4395        3857 :   return m;
    4396             : }
    4397             : 
    4398             : GEN
    4399         847 : conjclasses_algcenter(GEN cc, GEN p)
    4400             : {
    4401         847 :   GEN mt, elts = gel(cc,1), conjclass = gel(cc,2), rep = gel(cc,3), card;
    4402         847 :   long i, nbcl = lg(rep)-1, n = lg(elts)-1;
    4403             :   pari_sp av;
    4404             : 
    4405         847 :   card = zero_Flv(nbcl);
    4406       14819 :   for (i=1; i<=n; i++) card[conjclass[i]]++;
    4407             : 
    4408             :   /* multiplication table of the center of Z[G] (class functions) */
    4409         847 :   mt = cgetg(nbcl+1,t_VEC);
    4410        7217 :   for (i=1;i<=nbcl;i++) gel(mt,i) = zero_Flm_copy(nbcl,nbcl);
    4411         847 :   av = avma;
    4412        7217 :   for (i=1;i<=nbcl;i++)
    4413             :   {
    4414        6370 :     GEN xi = gel(elts,rep[i]), mi = gel(mt,i);
    4415             :     long j,k;
    4416      132244 :     for (j=1;j<=n;j++)
    4417             :     {
    4418      125874 :       GEN xj = gel(elts,j);
    4419      125874 :       k = vecsearch(elts, perm_mul(xi,xj), NULL);
    4420      125874 :       ucoeff(mi, conjclass[k], conjclass[j])++;
    4421             :     }
    4422       70238 :     for (k=1; k<=nbcl; k++)
    4423      852362 :       for (j=1; j<=nbcl; j++)
    4424             :       {
    4425      788494 :         ucoeff(mi,k,j) *= card[i];
    4426      788494 :         ucoeff(mi,k,j) /= card[k];
    4427             :       }
    4428        6370 :     set_avma(av);
    4429             :   }
    4430        7217 :   for (i=1;i<=nbcl;i++) gel(mt,i) = Flm_to_ZM(gel(mt,i));
    4431         847 :   return algtableinit_i(mt,p);
    4432             : }
    4433             : 
    4434             : GEN
    4435         329 : alggroupcenter(GEN G, GEN p, GEN *pcc)
    4436             : {
    4437         329 :   pari_sp av = avma;
    4438         329 :   GEN cc = group_to_cc(G), al = conjclasses_algcenter(cc, p);
    4439         315 :   if (!pcc) al = gerepilecopy(av,al);
    4440             :   else
    4441           7 :   { *pcc = cc; gerepileall(av,2,&al,pcc); }
    4442         315 :   return al;
    4443             : }
    4444             : 
    4445             : static GEN
    4446         294 : groupelts_algebra(GEN elts, GEN p)
    4447             : {
    4448         294 :   pari_sp av = avma;
    4449             :   GEN mt;
    4450         294 :   long i, n = lg(elts)-1;
    4451         294 :   elts = list_to_regular_rep(elts,n);
    4452         294 :   mt = cgetg(n+1, t_VEC);
    4453        4151 :   for (i=1; i<=n; i++) gel(mt,i) = matrix_perm(gel(elts,i),n);
    4454         294 :   return gerepilecopy(av, algtableinit_i(mt,p));
    4455             : }
    4456             : 
    4457             : GEN
    4458         329 : alggroup(GEN gal, GEN p)
    4459             : {
    4460         329 :   GEN elts = checkgroupelts(gal);
    4461         294 :   return groupelts_algebra(elts, p);
    4462             : }
    4463             : 
    4464             : /** MAXIMAL ORDER **/
    4465             : 
    4466             : GEN
    4467           0 : alg_changeorder(GEN al, GEN ord)
    4468             : {
    4469             :   GEN al2, mt, iord, mtx;
    4470             :   long i, n;
    4471           0 :   pari_sp av = avma;
    4472             : 
    4473           0 :   if (!gequal0(gel(al,10)))
    4474           0 :     pari_err_DOMAIN("alg_changeorder","characteristic","!=",gen_0,gel(al,10));
    4475           0 :   n = alg_get_absdim(al);
    4476             : 
    4477           0 :   iord = QM_inv(ord);
    4478           0 :   al2 = shallowcopy(al);
    4479             : 
    4480           0 :   gel(al2,7) = RgM_mul(gel(al2,7), ord);
    4481             : 
    4482           0 :   gel(al2,8) = RgM_mul(iord, gel(al,8));
    4483             : 
    4484           0 :   mt = cgetg(n+1,t_VEC);
    4485           0 :   gel(mt,1) = matid(n);
    4486           0 :   for (i=2; i<=n; i++) {
    4487           0 :     mtx = algbasismultable(al,gel(ord,i));
    4488           0 :     gel(mt,i) = RgM_mul(iord, RgM_mul(mtx, ord));
    4489             :   }
    4490           0 :   gel(al2,9) = mt;
    4491             : 
    4492           0 :   gel(al2,11) = algtracebasis(al2);
    4493             : 
    4494           0 :   return gerepilecopy(av,al2);
    4495             : }
    4496             : 
    4497             : static GEN
    4498       52955 : mattocol(GEN M, long n)
    4499             : {
    4500       52955 :   GEN C = cgetg(n*n+1, t_COL);
    4501             :   long i,j,ic;
    4502       52955 :   ic = 1;
    4503     1160880 :   for (i=1; i<=n; i++)
    4504    30025716 :   for (j=1; j<=n; j++, ic++) gel(C,ic) = gcoeff(M,i,j);
    4505       52955 :   return C;
    4506             : }
    4507             : 
    4508             : /* Ip is a lift of a left O/pO-ideal where O is the integral basis of al */
    4509             : static GEN
    4510        3738 : algleftordermodp(GEN al, GEN Ip, GEN p)
    4511             : {
    4512        3738 :   pari_sp av = avma;
    4513             :   GEN I, Ii, M, mt, K, imi, p2;
    4514             :   long n, i;
    4515        3738 :   n = alg_get_absdim(al);
    4516        3738 :   mt = alg_get_multable(al);
    4517        3738 :   p2 = sqri(p);
    4518             : 
    4519        3738 :   I = ZM_hnfmodid(Ip, p);
    4520        3738 :   Ii = ZM_inv(I,NULL);
    4521             : 
    4522        3738 :   M = cgetg(n+1, t_MAT);
    4523       56693 :   for (i=1; i<=n; i++) {
    4524       52955 :     imi = FpM_mul(Ii, FpM_mul(gel(mt,i), I, p2), p2);
    4525       52955 :     imi = ZM_Z_divexact(imi, p);
    4526       52955 :     gel(M,i) = mattocol(imi, n);
    4527             :   }
    4528        3738 :   K = FpM_ker(M, p);
    4529        3738 :   if (lg(K)==1) { set_avma(av); return matid(n); }
    4530        1701 :   K = ZM_hnfmodid(K,p);
    4531             : 
    4532        1701 :   return gerepileupto(av, ZM_Z_div(K,p));
    4533             : }
    4534             : 
    4535             : static GEN
    4536        4816 : alg_ordermodp(GEN al, GEN p)
    4537             : {
    4538             :   GEN alp;
    4539        4816 :   long i, N = alg_get_absdim(al);
    4540        4816 :   alp = cgetg(12, t_VEC);
    4541       43344 :   for (i=1; i<=8; i++) gel(alp,i) = gen_0;
    4542        4816 :   gel(alp,9) = cgetg(N+1, t_VEC);
    4543       58996 :   for (i=1; i<=N; i++) gmael(alp,9,i) = FpM_red(gmael(al,9,i), p);
    4544        4816 :   gel(alp,10) = p;
    4545        4816 :   gel(alp,11) = cgetg(N+1, t_VEC);
    4546       58996 :   for (i=1; i<=N; i++) gmael(alp,11,i) = Fp_red(gmael(al,11,i), p);
    4547             : 
    4548        4816 :   return alp;
    4549             : }
    4550             : 
    4551             : static GEN
    4552        2891 : algpradical_i(GEN al, GEN p, GEN zprad, GEN projs)
    4553             : {
    4554        2891 :   pari_sp av = avma;
    4555        2891 :   GEN alp = alg_ordermodp(al, p), liftrad, projrad, alq, alrad, res, Lalp, radq;
    4556             :   long i;
    4557        2891 :   if (lg(zprad)==1) {
    4558        1806 :     liftrad = NULL;
    4559        1806 :     projrad = NULL;
    4560             :   }
    4561             :   else {
    4562        1085 :     alq = alg_quotient(alp, zprad, 1);
    4563        1085 :     alp = gel(alq,1);
    4564        1085 :     projrad = gel(alq,2);
    4565        1085 :     liftrad = gel(alq,3);
    4566             :   }
    4567             : 
    4568        2891 :   if (projs) {
    4569         378 :     if (projrad) {
    4570          28 :       projs = gcopy(projs);
    4571          84 :       for (i=1; i<lg(projs); i++)
    4572          56 :         gel(projs,i) = FpM_FpC_mul(projrad, gel(projs,i), p);
    4573             :     }
    4574         378 :     Lalp = alg_centralproj(alp, projs, 1);
    4575             : 
    4576         378 :     alrad = cgetg(lg(Lalp),t_VEC);
    4577        1687 :     for (i=1; i<lg(Lalp); i++) {
    4578        1309 :       alq = gel(Lalp,i);
    4579        1309 :       radq = algradical(gel(alq,1));
    4580        1309 :       if (gequal0(radq))
    4581         847 :         gel(alrad,i) = cgetg(1,t_MAT);
    4582             :       else {
    4583         462 :         radq = FpM_mul(gel(alq,3),radq,p);
    4584         462 :         gel(alrad,i) = radq;
    4585             :       }
    4586             :     }
    4587         378 :     alrad = shallowmatconcat(alrad);
    4588         378 :     alrad = FpM_image(alrad,p);
    4589             :   }
    4590        2513 :   else alrad = algradical(alp);
    4591             : 
    4592        2891 :   if (!gequal0(alrad)) {
    4593        2219 :     if (liftrad) alrad = FpM_mul(liftrad, alrad, p);
    4594        2219 :     res = shallowmatconcat(mkvec2(alrad, zprad));
    4595        2219 :     res = FpM_image(res,p);
    4596             :   }
    4597         672 :   else res = lg(zprad)==1 ? gen_0 : zprad;
    4598        2891 :   return gerepilecopy(av, res);
    4599             : }
    4600             : 
    4601             : static GEN
    4602        1925 : algpdecompose0(GEN al, GEN prad, GEN p, GEN projs)
    4603             : {
    4604        1925 :   pari_sp av = avma;
    4605        1925 :   GEN alp, quo, ss, liftm = NULL, projm = NULL, dec, res, I, Lss, deci;
    4606             :   long i, j;
    4607             : 
    4608        1925 :   alp = alg_ordermodp(al, p);
    4609        1925 :   if (!gequal0(prad)) {
    4610        1519 :     quo = alg_quotient(alp, prad, 1);
    4611        1519 :     ss = gel(quo,1);
    4612        1519 :     projm = gel(quo,2);
    4613        1519 :     liftm = gel(quo,3);
    4614             :   }
    4615         406 :   else ss = alp;
    4616             : 
    4617        1925 :   if (projs) {
    4618         322 :     if (projm) {
    4619        1029 :       for (i=1; i<lg(projs); i++)
    4620         784 :         gel(projs,i) = FpM_FpC_mul(projm, gel(projs,i), p);
    4621             :     }
    4622         322 :     Lss = alg_centralproj(ss, projs, 1);
    4623             : 
    4624         322 :     dec = cgetg(lg(Lss),t_VEC);
    4625        1512 :     for (i=1; i<lg(Lss); i++) {
    4626        1190 :       gel(dec,i) = algsimpledec_ss(gmael(Lss,i,1), 1);
    4627        1190 :       deci = gel(dec,i);
    4628        2625 :       for (j=1; j<lg(deci); j++)
    4629        1435 :        gmael(deci,j,3) = FpM_mul(gmael(Lss,i,3), gmael(deci,j,3), p);
    4630             :     }
    4631         322 :     dec = shallowconcat1(dec);
    4632             :   }
    4633        1603 :   else dec = algsimpledec_ss(ss,1);
    4634             : 
    4635        1925 :   res = cgetg(lg(dec),t_VEC);
    4636        5670 :   for (i=1; i<lg(dec); i++) {
    4637        3745 :     I = gmael(dec,i,3);
    4638        3745 :     if (liftm) I = FpM_mul(liftm,I,p);
    4639        3745 :     I = shallowmatconcat(mkvec2(I,prad));
    4640        3745 :     gel(res,i) = I;
    4641             :   }
    4642             : 
    4643        1925 :   return gerepilecopy(av, res);
    4644             : }
    4645             : 
    4646             : /* finds a nontrivial ideal of O/prad or gen_0 if there is none. */
    4647             : static GEN
    4648         735 : algpdecompose_i(GEN al, GEN p, GEN zprad, GEN projs)
    4649             : {
    4650         735 :   pari_sp av = avma;
    4651         735 :   GEN prad = algpradical_i(al,p,zprad,projs);
    4652         735 :   return gerepileupto(av, algpdecompose0(al, prad, p, projs));
    4653             : }
    4654             : 
    4655             : /* ord is assumed to be in hnf wrt the integral basis of al. */
    4656             : /* assumes that alg_get_invbasis(al) is integral. */
    4657             : static GEN
    4658        1701 : alg_change_overorder_shallow(GEN al, GEN ord)
    4659             : {
    4660             :   GEN al2, mt, iord, mtx, den, den2, div;
    4661             :   long i, n;
    4662        1701 :   n = alg_get_absdim(al);
    4663             : 
    4664        1701 :   iord = QM_inv(ord);
    4665        1701 :   al2 = shallowcopy(al);
    4666        1701 :   ord = Q_remove_denom(ord,&den);
    4667             : 
    4668        1701 :   gel(al2,7) = Q_remove_denom(gel(al,7), &den2);
    4669        1701 :   if (den2) div = mulii(den,den2);
    4670         644 :   else      div = den;
    4671        1701 :   gel(al2,7) = ZM_Z_div(ZM_mul(gel(al2,7), ord), div);
    4672             : 
    4673        1701 :   gel(al2,8) = ZM_mul(iord, gel(al,8));
    4674             : 
    4675        1701 :   mt = cgetg(n+1,t_VEC);
    4676        1701 :   gel(mt,1) = matid(n);
    4677        1701 :   div = sqri(den);
    4678       19418 :   for (i=2; i<=n; i++) {
    4679       17717 :     mtx = algbasismultable(al,gel(ord,i));
    4680       17717 :     gel(mt,i) = ZM_mul(iord, ZM_mul(mtx, ord));
    4681       17717 :     gel(mt,i) = ZM_Z_divexact(gel(mt,i), div);
    4682             :   }
    4683        1701 :   gel(al2,9) = mt;
    4684             : 
    4685        1701 :   gel(al2,11) = algtracebasis(al2);
    4686             : 
    4687        1701 :   return al2;
    4688             : }
    4689             : 
    4690             : static GEN
    4691       10255 : algfromcenter(GEN al, GEN x)
    4692             : {
    4693       10255 :   GEN nf = alg_get_center(al);
    4694             :   long n;
    4695       10255 :   switch(alg_type(al)) {
    4696        9163 :     case al_CYCLIC:
    4697        9163 :       n = alg_get_degree(al);
    4698        9163 :       break;
    4699        1092 :     case al_CSA:
    4700        1092 :       n = alg_get_dim(al);
    4701        1092 :       break;
    4702           0 :     default:
    4703             :       return NULL; /*LCOV_EXCL_LINE*/
    4704             :   }
    4705       10255 :   return algalgtobasis(al, scalarcol(basistoalg(nf, x), n));
    4706             : }
    4707             : 
    4708             : /* x is an ideal of the center in hnf form */
    4709             : static GEN
    4710        2891 : algfromcenterhnf(GEN al, GEN x)
    4711             : {
    4712             :   GEN res;
    4713             :   long i;
    4714        2891 :   res = cgetg(lg(x), t_MAT);
    4715        9324 :   for (i=1; i<lg(x); i++) gel(res,i) = algfromcenter(al, gel(x,i));
    4716        2891 :   return res;
    4717             : }
    4718             : 
    4719             : /* assumes al is CSA or CYCLIC */
    4720             : static GEN
    4721        1190 : algcenter_precompute(GEN al, GEN p)
    4722             : {
    4723        1190 :   GEN fa, pdec, nfprad, projs, nf = alg_get_center(al);
    4724             :   long i, np;
    4725             : 
    4726        1190 :   pdec = idealprimedec(nf, p);
    4727        1190 :   settyp(pdec, t_COL);
    4728        1190 :   np = lg(pdec)-1;
    4729        1190 :   fa = mkmat2(pdec, const_col(np, gen_1));
    4730        1190 :   if (dvdii(nf_get_disc(nf), p))
    4731         329 :     nfprad = idealprodprime(nf, pdec);
    4732             :   else
    4733         861 :     nfprad = scalarmat_shallow(p, nf_get_degree(nf));
    4734        1190 :   fa = idealchineseinit(nf, fa);
    4735        1190 :   projs = cgetg(np+1, t_VEC);
    4736        2835 :   for (i=1; i<=np; i++) gel(projs, i) = idealchinese(nf, fa, vec_ei(np,i));
    4737        1190 :   return mkvec2(nfprad, projs);
    4738             : }
    4739             : 
    4740             : static GEN
    4741        2891 : algcenter_prad(GEN al, GEN p, GEN pre)
    4742             : {
    4743             :   GEN nfprad, zprad, mtprad;
    4744             :   long i;
    4745        2891 :   nfprad = gel(pre,1);
    4746        2891 :   zprad = algfromcenterhnf(al, nfprad);
    4747        2891 :   zprad = FpM_image(zprad, p);
    4748        2891 :   mtprad = cgetg(lg(zprad), t_VEC);
    4749        4578 :   for (i=1; i<lg(zprad); i++) gel(mtprad, i) = algbasismultable(al, gel(zprad,i));
    4750        2891 :   mtprad = shallowmatconcat(mtprad);
    4751        2891 :   zprad = FpM_image(mtprad, p);
    4752        2891 :   return zprad;
    4753             : }
    4754             : 
    4755             : static GEN
    4756        2891 : algcenter_p_projs(GEN al, GEN p, GEN pre)
    4757             : {
    4758             :   GEN projs, zprojs;
    4759             :   long i;
    4760        2891 :   projs = gel(pre,2);
    4761        2891 :   zprojs = cgetg(lg(projs), t_VEC);
    4762        6713 :   for (i=1; i<lg(projs); i++) gel(zprojs,i) = FpC_red(algfromcenter(al, gel(projs,i)),p);
    4763        2891 :   return zprojs;
    4764             : }
    4765             : 
    4766             : /* al is assumed to be simple */
    4767             : static GEN
    4768        1190 : alg_pmaximal(GEN al, GEN p)
    4769             : {
    4770        1190 :   GEN al2, prad, lord = gen_0, I, id, dec, zprad, projs, pre;
    4771             :   long n, i;
    4772        1190 :   n = alg_get_absdim(al);
    4773        1190 :   id = matid(n);
    4774        1190 :   al2 = al;
    4775             : 
    4776        1190 :   dbg_printf(0)("Round 2 (non-commutative) at p=%Ps, dim=%d\n", p, n);
    4777             : 
    4778        1190 :   pre = algcenter_precompute(al,p);
    4779             : 
    4780             :   while (1) {
    4781        2156 :     zprad = algcenter_prad(al2, p, pre);
    4782        2156 :     projs = algcenter_p_projs(al2, p, pre);
    4783        2156 :     if (lg(projs) == 2) projs = NULL;
    4784        2156 :     prad = algpradical_i(al2,p,zprad,projs);
    4785        2156 :     if (typ(prad) == t_INT) break;
    4786        2135 :     lord = algleftordermodp(al2,prad,p);
    4787        2135 :     if (!cmp_universal(lord,id)) break;
    4788         966 :     al2 = alg_change_overorder_shallow(al2,lord);
    4789             :   }
    4790             : 
    4791        1190 :   dec = algpdecompose0(al2,prad,p,projs);
    4792        1925 :   while (lg(dec)>2) {
    4793        1799 :     for (i=1; i<lg(dec); i++) {
    4794        1603 :       I = gel(dec,i);
    4795        1603 :       lord = algleftordermodp(al2,I,p);
    4796        1603 :       if (cmp_universal(lord,matid(n))) break;
    4797             :     }
    4798         931 :     if (i==lg(dec)) break;
    4799         735 :     al2 = alg_change_overorder_shallow(al2,lord);
    4800         735 :     zprad = algcenter_prad(al2, p, pre);
    4801         735 :     projs = algcenter_p_projs(al2, p, pre);
    4802         735 :     if (lg(projs) == 2) projs = NULL;
    4803         735 :     dec = algpdecompose_i(al2,p,zprad,projs);
    4804             :   }
    4805        1190 :   return al2;
    4806             : }
    4807             : 
    4808             : static GEN
    4809        5446 : algtracematrix(GEN al)
    4810             : {
    4811             :   GEN M, mt;
    4812             :   long n, i, j;
    4813        5446 :   n = alg_get_absdim(al);
    4814        5446 :   mt = alg_get_multable(al);
    4815        5446 :   M = cgetg(n+1, t_MAT);
    4816       43155 :   for (i=1; i<=n; i++)
    4817             :   {
    4818       37709 :     gel(M,i) = cgetg(n+1,t_MAT);
    4819      273945 :     for (j=1; j<=i; j++)
    4820      236236 :       gcoeff(M,j,i) = gcoeff(M,i,j) = algabstrace(al,gmael(mt,i,j));
    4821             :   }
    4822        5446 :   return M;
    4823             : }
    4824             : static GEN
    4825         133 : algdisc_i(GEN al) { return ZM_det(algtracematrix(al)); }
    4826             : GEN
    4827           7 : algdisc(GEN al)
    4828             : {
    4829           7 :   pari_sp av = avma;
    4830           7 :   checkalg(al); return gerepileuptoint(av, algdisc_i(al));
    4831             : }
    4832             : static GEN
    4833         126 : alg_maximal(GEN al)
    4834             : {
    4835         126 :   GEN fa = absZ_factor(algdisc_i(al));
    4836         126 :   return alg_maximal_primes(al, gel(fa,1));
    4837             : }
    4838             : 
    4839             : /** LATTICES **/
    4840             : 
    4841             : /*
    4842             :  Convention: lattice = [I,t] representing t*I, where
    4843             :  - I integral nonsingular upper-triangular matrix representing a lattice over
    4844             :    the integral basis of the algebra, and
    4845             :  - t>0 either an integer or a rational number.
    4846             : 
    4847             :  Recommended and returned by the functions below:
    4848             :  - I HNF and primitive
    4849             : */
    4850             : 
    4851             : /* TODO use hnfmodid whenever possible using a*O <= I <= O
    4852             :  * for instance a = ZM_det_triangular(I) */
    4853             : 
    4854             : static GEN
    4855       63343 : primlat(GEN lat)
    4856             : {
    4857             :   GEN m, t, c;
    4858       63343 :   m = alglat_get_primbasis(lat);
    4859       63343 :   t = alglat_get_scalar(lat);
    4860       63343 :   m = Q_primitive_part(m,&c);
    4861       63343 :   if (c) return mkvec2(m,gmul(t,c));
    4862       53760 :   return lat;
    4863             : }
    4864             : 
    4865             : /* assumes the lattice contains d * integral basis, d=0 allowed */
    4866             : GEN
    4867       51065 : alglathnf(GEN al, GEN m, GEN d)
    4868             : {
    4869       51065 :   pari_sp av = avma;
    4870             :   long N,i,j;
    4871             :   GEN m2, c;
    4872       51065 :   checkalg(al);
    4873       51065 :   N = alg_get_absdim(al);
    4874       51065 :   if (!d) d = gen_0;
    4875       51065 :   if (typ(m) == t_VEC) m = matconcat(m);
    4876       51065 :   if (typ(m) == t_COL) m = algleftmultable(al,m);
    4877       51065 :   if (typ(m) != t_MAT) pari_err_TYPE("alglathnf",m);
    4878       51058 :   if (typ(d) != t_FRAC && typ(d) != t_INT) pari_err_TYPE("alglathnf",d);
    4879       51058 :   if (lg(m)-1 < N || lg(gel(m,1))-1 != N) pari_err_DIM("alglathnf");
    4880      459242 :   for (i=1; i<=N; i++)
    4881     6820758 :     for (j=1; j<lg(m); j++)
    4882     6412546 :       if (typ(gcoeff(m,i,j)) != t_FRAC && typ(gcoeff(m,i,j)) != t_INT)
    4883           7 :         pari_err_TYPE("alglathnf", gcoeff(m,i,j));
    4884       51023 :   m2 = Q_primitive_part(m,&c);
    4885       51023 :   if (!c) c = gen_1;
    4886       51023 :   if (!signe(d)) d = detint(m2);
    4887       45593 :   else           d = gdiv(d,c); /* should be an integer */
    4888       51023 :   if (!signe(d)) pari_err_INV("alglathnf [m does not have full rank]", m2);
    4889       51009 :   m2 = ZM_hnfmodid(m2,d);
    4890       51009 :   return gerepilecopy(av, mkvec2(m2,c));
    4891             : }
    4892             : 
    4893             : static GEN
    4894       10689 : prepare_multipliers(GEN *a, GEN *b)
    4895             : {
    4896             :   GEN na, nb, da, db, d;
    4897       10689 :   na = numer_i(*a); da = denom_i(*a);
    4898       10689 :   nb = numer_i(*b); db = denom_i(*b);
    4899       10689 :   na = mulii(na,db);
    4900       10689 :   nb = mulii(nb,da);
    4901       10689 :   d = gcdii(na,nb);
    4902       10689 :   *a = diviiexact(na,d);
    4903       10689 :   *b = diviiexact(nb,d);
    4904       10689 :   return gdiv(d, mulii(da,db));
    4905             : }
    4906             : 
    4907             : static GEN
    4908       10689 : prepare_lat(GEN m1, GEN t1, GEN m2, GEN t2)
    4909             : {
    4910       10689 :   GEN d = prepare_multipliers(&t1, &t2);
    4911       10689 :   m1 = ZM_Z_mul(m1,t1);
    4912       10689 :   m2 = ZM_Z_mul(m2,t2);
    4913       10689 :   return mkvec3(m1,m2,d);
    4914             : }
    4915             : 
    4916             : static GEN
    4917       10689 : alglataddinter(GEN al, GEN lat1, GEN lat2, GEN *sum, GEN *inter)
    4918             : {
    4919             :   GEN d, m1, m2, t1, t2, M, prep, d1, d2, ds, di, K;
    4920       10689 :   checkalg(al);
    4921       10689 :   checklat(al,lat1);
    4922       10689 :   checklat(al,lat2);
    4923             : 
    4924       10689 :   m1 = alglat_get_primbasis(lat1);
    4925       10689 :   t1 = alglat_get_scalar(lat1);
    4926       10689 :   m2 = alglat_get_primbasis(lat2);
    4927       10689 :   t2 = alglat_get_scalar(lat2);
    4928       10689 :   prep = prepare_lat(m1, t1, m2, t2);
    4929       10689 :   m1 = gel(prep,1);
    4930       10689 :   m2 = gel(prep,2);
    4931       10689 :   d = gel(prep,3);
    4932       10689 :   M = matconcat(mkvec2(m1,m2));
    4933       10689 :   d1 = ZM_det_triangular(m1);
    4934       10689 :   d2 = ZM_det_triangular(m2);
    4935       10689 :   ds = gcdii(d1,d2);
    4936       10689 :   if (inter)
    4937             :   {
    4938        7112 :     di = diviiexact(mulii(d1,d2),ds);
    4939        7112 :     K = matkermod(M,di,sum);
    4940        7112 :     K = rowslice(K,1,lg(m1));
    4941        7112 :     *inter = hnfmodid(FpM_mul(m1,K,di),di);
    4942        7112 :     if (sum) *sum = hnfmodid(*sum,ds);
    4943             :   }
    4944        3577 :   else *sum = hnfmodid(M,ds);
    4945       10689 :   return d;
    4946             : }
    4947             : 
    4948             : GEN
    4949        3598 : alglatinter(GEN al, GEN lat1, GEN lat2, GEN* ptsum)
    4950             : {
    4951        3598 :   pari_sp av = avma;
    4952             :   GEN inter, d;
    4953        3598 :   d = alglataddinter(al, lat1, lat2, ptsum, &inter);
    4954        3598 :   inter = primlat(mkvec2(inter, d));
    4955        3598 :   if (ptsum)
    4956             :   {
    4957          14 :     *ptsum = primlat(mkvec2(*ptsum,d));
    4958          14 :     gerepileall(av, 2, &inter, ptsum);
    4959             :   }
    4960        3584 :   else inter = gerepilecopy(av, inter);
    4961        3598 :   return inter;
    4962             : }
    4963             : 
    4964             : GEN
    4965        7091 : alglatadd(GEN al, GEN lat1, GEN lat2, GEN* ptinter)
    4966             : {
    4967        7091 :   pari_sp av = avma;
    4968             :   GEN sum, d;
    4969        7091 :   d = alglataddinter(al, lat1, lat2, &sum, ptinter);
    4970        7091 :   sum = primlat(mkvec2(sum, d));
    4971        7091 :   if (ptinter)
    4972             :   {
    4973        3514 :     *ptinter = primlat(mkvec2(*ptinter,d));
    4974        3514 :     gerepileall(av, 2, &sum, ptinter);
    4975             :   }
    4976        3577 :   else sum = gerepilecopy(av, sum);
    4977        7091 :   return sum;
    4978             : }
    4979             : 
    4980             : int
    4981       31549 : alglatsubset(GEN al, GEN lat1, GEN lat2, GEN* ptindex)
    4982             : {
    4983             :   /* TODO version that returns the quotient as abelian group? */
    4984             :   /* return matrices to convert coordinates from one to other? */
    4985       31549 :   pari_sp av = avma;
    4986             :   int res;
    4987             :   GEN m1, m2, m2i, m, t;
    4988       31549 :   checkalg(al);
    4989       31549 :   checklat(al,lat1);
    4990       31549 :   checklat(al,lat2);
    4991       31549 :   m1 = alglat_get_primbasis(lat1);
    4992       31549 :   m2 = alglat_get_primbasis(lat2);
    4993       31549 :   m2i = RgM_inv_upper(m2);
    4994       31549 :   t = gdiv(alglat_get_scalar(lat1), alglat_get_scalar(lat2));
    4995       31549 :   m = RgM_Rg_mul(RgM_mul(m2i,m1), t);
    4996       31549 :   res = RgM_is_ZM(m);
    4997       31549 :   if (res && ptindex)
    4998             :   {
    4999        1757 :     *ptindex = mpabs(ZM_det_triangular(m));
    5000        1757 :     gerepileall(av,1,ptindex);
    5001             :   }
    5002       29792 :   else set_avma(av);
    5003       31549 :   return res;
    5004             : }
    5005             : 
    5006             : GEN
    5007        5264 : alglatindex(GEN al, GEN lat1, GEN lat2)
    5008             : {
    5009        5264 :   pari_sp av = avma;
    5010             :   long N;
    5011             :   GEN res;
    5012        5264 :   checkalg(al);
    5013        5264 :   checklat(al,lat1);
    5014        5264 :   checklat(al,lat2);
    5015        5264 :   N = alg_get_absdim(al);
    5016        5264 :   res = alglat_get_scalar(lat1);
    5017        5264 :   res = gdiv(res, alglat_get_scalar(lat2));
    5018        5264 :   res = gpowgs(res, N);
    5019        5264 :   res = gmul(res,RgM_det_triangular(alglat_get_primbasis(lat1)));
    5020        5264 :   res = gdiv(res, RgM_det_triangular(alglat_get_primbasis(lat2)));
    5021        5264 :   res = gabs(res,0);
    5022        5264 :   return gerepilecopy(av, res);
    5023             : }
    5024             : 
    5025             : GEN
    5026       45605 : alglatmul(GEN al, GEN lat1, GEN lat2)
    5027             : {
    5028       45605 :   pari_sp av = avma;
    5029             :   long N,i;
    5030             :   GEN m1, m2, m, V, lat, t, d, dp;
    5031       45605 :   checkalg(al);
    5032       45605 :   if (typ(lat1)==t_COL)
    5033             :   {
    5034       19292 :     if (typ(lat2)==t_COL)
    5035           7 :       pari_err_TYPE("alglatmul [one of lat1, lat2 has to be a lattice]", lat2);
    5036       19285 :     checklat(al,lat2);
    5037       19285 :     lat1 = Q_remove_denom(lat1,&d);
    5038       19285 :     m = algbasismultable(al,lat1);
    5039       19285 :     m2 = alglat_get_primbasis(lat2);
    5040       19285 :     dp = mulii(detint(m),ZM_det_triangular(m2));
    5041       19285 :     m = ZM_mul(m,m2);
    5042       19285 :     t = alglat_get_scalar(lat2);
    5043       19285 :     if (d) t = gdiv(t,d);
    5044             :   }
    5045             :   else /* typ(lat1)!=t_COL */
    5046             :   {
    5047       26313 :     checklat(al,lat1);
    5048       26313 :     if (typ(lat2)==t_COL)
    5049             :     {
    5050       19285 :       lat2 = Q_remove_denom(lat2,&d);
    5051       19285 :       m = algbasisrightmultable(al,lat2);
    5052       19285 :       m1 = alglat_get_primbasis(lat1);
    5053       19285 :       dp = mulii(detint(m),ZM_det_triangular(m1));
    5054       19285 :       m = ZM_mul(m,m1);
    5055       19285 :       t = alglat_get_scalar(lat1);
    5056       19285 :       if (d) t = gdiv(t,d);
    5057             :     }
    5058             :     else /* typ(lat2)!=t_COL */
    5059             :     {
    5060        7028 :       checklat(al,lat2);
    5061        7021 :       N = alg_get_absdim(al);
    5062        7021 :       m1 = alglat_get_primbasis(lat1);
    5063        7021 :       m2 = alglat_get_primbasis(lat2);
    5064        7021 :       dp = mulii(ZM_det_triangular(m1), ZM_det_triangular(m2));
    5065        7021 :       V = cgetg(N+1,t_VEC);
    5066       63189 :       for (i=1; i<=N; i++) {
    5067       56168 :         gel(V,i) = algbasismultable(al,gel(m1,i));
    5068       56168 :         gel(V,i) = ZM_mul(gel(V,i),m2);
    5069             :       }
    5070        7021 :       m = matconcat(V);
    5071        7021 :       t = gmul(alglat_get_scalar(lat1), alglat_get_scalar(lat2));
    5072             :     }
    5073             :   }
    5074             : 
    5075       45591 :   lat = alglathnf(al,m,dp);
    5076       45591 :   gel(lat,2) = gmul(alglat_get_scalar(lat), t);
    5077       45591 :   lat = primlat(lat);
    5078       45591 :   return gerepilecopy(av, lat);
    5079             : }
    5080             : 
    5081             : int
    5082       17521 : alglatcontains(GEN al, GEN lat, GEN x, GEN *ptc)
    5083             : {
    5084       17521 :   pari_sp av = avma;
    5085             :   GEN m, t, sol;
    5086       17521 :   checkalg(al);
    5087       17521 :   checklat(al,lat);
    5088       17521 :   m = alglat_get_primbasis(lat);
    5089       17521 :   t = alglat_get_scalar(lat);
    5090       17521 :   x = RgC_Rg_div(x,t);
    5091       17521 :   if (!RgV_is_ZV(x)) return gc_bool(av,0);
    5092       17521 :   sol = hnf_solve(m,x);
    5093       17521 :   if (!sol) return gc_bool(av,0);
    5094        8771 :   if (!ptc) return gc_bool(av,1);
    5095        8764 :   *ptc = sol; gerepileall(av,1,ptc); return 1;
    5096             : }
    5097             : 
    5098             : GEN
    5099        8771 : alglatelement(GEN al, GEN lat, GEN c)
    5100             : {
    5101        8771 :   pari_sp av = avma;
    5102             :   GEN res;
    5103        8771 :   checkalg(al);
    5104        8771 :   checklat(al,lat);
    5105        8771 :   if (typ(c)!=t_COL) pari_err_TYPE("alglatelement", c);
    5106        8764 :   res = ZM_ZC_mul(alglat_get_primbasis(lat),c);
    5107        8764 :   res = RgC_Rg_mul(res, alglat_get_scalar(lat));
    5108        8764 :   return gerepilecopy(av,res);
    5109             : }
    5110             : 
    5111             : /* idem QM_invimZ, knowing result is contained in 1/c*Z^n */
    5112             : static GEN
    5113        3535 : QM_invimZ_mod(GEN m, GEN c)
    5114             : {
    5115             :   GEN d, m0, K;
    5116        3535 :   m0 = Q_remove_denom(m, &d);
    5117        3535 :   if (d)    d = mulii(d,c);
    5118          21 :   else      d = c;
    5119        3535 :   K = matkermod(m0, d, NULL);
    5120        3535 :   if (lg(K)==1) K = scalarmat(d, lg(m)-1);
    5121        3521 :   else          K = hnfmodid(K, d);
    5122        3535 :   return RgM_Rg_div(K,c);
    5123             : }
    5124             : 
    5125             : /* If m is injective, computes a Z-basis of the submodule of elements whose
    5126             :  * image under m is integral */
    5127             : static GEN
    5128          14 : QM_invimZ(GEN m)
    5129             : {
    5130          14 :   return RgM_invimage(m, QM_ImQ_hnf(m));
    5131             : }
    5132             : 
    5133             : /* An isomorphism of R-modules M_{m,n}(R) -> R^{m*n} */
    5134             : static GEN
    5135       28322 : mat2col(GEN M, long m, long n)
    5136             : {
    5137             :   long i,j,k,p;
    5138             :   GEN C;
    5139       28322 :   p = m*n;
    5140       28322 :   C = cgetg(p+1,t_COL);
    5141      254702 :   for (i=1,k=1;i<=m;i++)
    5142     2036804 :     for (j=1;j<=n;j++,k++)
    5143     1810424 :       gel(C,k) = gcoeff(M,i,j);
    5144       28322 :   return C;
    5145             : }
    5146             : 
    5147             : static GEN
    5148        3535 : alglattransporter_i(GEN al, GEN lat1, GEN lat2, long right)
    5149             : {
    5150             :   GEN m1, m2, m2i, M, MT, mt, t1, t2, T, c;
    5151             :   long N, i;
    5152        3535 :   N = alg_get_absdim(al);
    5153        3535 :   m1 = alglat_get_primbasis(lat1);
    5154        3535 :   m2 = alglat_get_primbasis(lat2);
    5155        3535 :   m2i = RgM_inv_upper(m2);
    5156        3535 :   c = detint(m1);
    5157        3535 :   t1 = alglat_get_scalar(lat1);
    5158        3535 :   m1 = RgM_Rg_mul(m1,t1);
    5159        3535 :   t2 = alglat_get_scalar(lat2);
    5160        3535 :   m2i = RgM_Rg_div(m2i,t2);
    5161             : 
    5162        3535 :   MT = right? NULL: alg_get_multable(al);
    5163        3535 :   M = cgetg(N+1, t_MAT);
    5164       31815 :   for (i=1; i<=N; i++) {
    5165       28280 :     if (right) mt = algbasisrightmultable(al, vec_ei(N,i));
    5166       14168 :     else       mt = gel(MT,i);
    5167       28280 :     mt = RgM_mul(m2i,mt);
    5168       28280 :     mt = RgM_mul(mt,m1);
    5169       28280 :     gel(M,i) = mat2col(mt, N, N);
    5170             :   }
    5171             : 
    5172        3535 :   c = gdiv(t2,gmul(c,t1));
    5173        3535 :   c = denom_i(c);
    5174        3535 :   T = QM_invimZ_mod(M,c);
    5175        3535 :   return primlat(mkvec2(T,gen_1));
    5176             : }
    5177             : 
    5178             : /*
    5179             :    { x in al | x*lat1 subset lat2}
    5180             : */
    5181             : GEN
    5182        1771 : alglatlefttransporter(GEN al, GEN lat1, GEN lat2)
    5183             : {
    5184        1771 :   pari_sp av = avma;
    5185        1771 :   checkalg(al);
    5186        1771 :   checklat(al,lat1);
    5187        1771 :   checklat(al,lat2);
    5188        1771 :   return gerepilecopy(av, alglattransporter_i(al,lat1,lat2,0));
    5189             : }
    5190             : 
    5191             : /*
    5192             :    { x in al | lat1*x subset lat2}
    5193             : */
    5194             : GEN
    5195        1764 : alglatrighttransporter(GEN al, GEN lat1, GEN lat2)
    5196             : {
    5197        1764 :   pari_sp av = avma;
    5198        1764 :   checkalg(al);
    5199        1764 :   checklat(al,lat1);
    5200        1764 :   checklat(al,lat2);
    5201        1764 :   return gerepilecopy(av, alglattransporter_i(al,lat1,lat2,1));
    5202             : }
    5203             : 
    5204             : GEN
    5205          42 : algmakeintegral(GEN mt0, long maps)
    5206             : {
    5207          42 :   pari_sp av = avma;
    5208             :   long n,i;
    5209             :   GEN m,P,Pi,mt2,mt;
    5210          42 :   n = lg(mt0)-1;
    5211          42 :   mt = check_mt(mt0,NULL);
    5212          42 :   if (!mt) pari_err_TYPE("algmakeintegral", mt0);
    5213          21 :   if (isint1(Q_denom(mt0))) {
    5214           7 :     if (maps) mt = mkvec3(mt,matid(n),matid(n));
    5215           7 :     return gerepilecopy(av,mt);
    5216             :   }
    5217          14 :   dbg_printf(2)(" algmakeintegral: dim=%d, denom=%Ps\n", n, Q_denom(mt0));
    5218          14 :   m = cgetg(n+1,t_MAT);
    5219          56 :   for (i=1;i<=n;i++)
    5220          42 :     gel(m,i) = mat2col(gel(mt,i),n,n);
    5221          14 :   dbg_printf(2)(" computing order, dims m = %d x %d...\n", nbrows(m), lg(m)-1);
    5222          14 :   P = QM_invimZ(m);
    5223          14 :   dbg_printf(2)(" ...done.\n");
    5224          14 :   P = shallowmatconcat(mkvec2(col_ei(n,1),P));
    5225          14 :   P = hnf(P);
    5226          14 :   Pi = RgM_inv(P);
    5227          14 :   mt2 = change_Rgmultable(mt,P,Pi);
    5228          14 :   if (maps) mt2 = mkvec3(mt2,Pi,P); /* mt2, mt->mt2, mt2->mt */
    5229          14 :   return gerepilecopy(av,mt2);
    5230             : }
    5231             : 
    5232             : /** ORDERS **/
    5233             : 
    5234             : /** IDEALS **/
    5235             : 

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