Re: size of the coefficients returned by bnfisnorm()

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

Le Sat, Oct 30, 2021 at 10:12:07PM -0400, Max Alekseyev a écrit :
> Dear Bill,
> 
> I did not have a chance to thank you for your suggestion on reducing
> coefficients of bnfisnorm() based on qfparam(), but now I have a similar
> question about qfparam() itself.
> Consider an example:
> 
> ? G = matdiagonal([650, -104329, -104329]);
> ? M = qfparam(G, qfsolve(G))
> %1 =
> [-33698267 -161709950 -194002198]
> [  -521645   -2487100   -2964370]
> [ -2608225  -12519480  -15023350]
> 
> I claim that the following matrix works equally well (i.e. it could have
> been returned by qfparam), but it has much smaller entries:
> 
> ? M2 = [323, 0, 323; 5, 50, -5; 25,- 10, - 25]

As I understand, your solution is not a full parametrization since it
does not reach all the rational solutions:

s=[-194002198,-2964370,-15023350]~
is a solution which is not reached by M2 over Q, only over Q(i).

? M2^-1*s
%36 = [-600625,775,-1]~

The first and last components must be squares so that
[x^2,x*y,y^2] = [-600625,775,-1] had a solution.

Cheers,
Bill.