Re: h_x of points on a rank-11 elliptic curve

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

On Fri, May 03, 2024 at 01:54:39PM +0400, Kevin Acres wrote:
> I’m interested in what games you can play with 107122676734733201

If one goes to the OEIS, one finds this page that you conveniently filled:
https://oeis.org/A173795

which tell us this is related to the equation

y^2 + x^3 = 107122676734733201

which by setting X=-x, Y=y can be written as

Y^2 = X^3+107122676734733201

So we can try to compute its Mordell-Weil group:

E=ellinit([0,107122676734733201]);
R=ellrank(E)
%7 = [12,12,0,[[201652,339591753],[13704427,50734146678],[33568399,194489678280],[-218856237137/567009,95104786928876776/426957777],[-609969146/5625,137253737419433/421875],[366673/4,2627765241/8],[7030108/9,20628557171/27],[15840274/25,75155457393/125],[633111217/16,15930169162647/64],[3710170714/24025,1239583908216063/3723875],[5405262670/5329,417296243200533/389017],[17169987767/3481,2250861276375248/205379]]]

So this is a rank 12 curve. by setting
Pvec=R[4];
the same trick applies:
M=Mat(apply(P->my([X,Y]=P);[X*Y,-X^2,Y,-X,-1],Pvec)~);
V=apply(P->my([X,Y]=P);X^3-Y^2,Pvec)~;
matsolve(M,V)
%11 = [0,0,0,0,107122676734733201]~

which is not surprising.

Cheers,
Bill.