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 12:04:54AM -0400, Charles Greathouse wrote:
> I'm trying to work my way through the paper
> https://arxiv.org/abs/2403.17955
> and I'm at Proposition 2.1.
> 
> I have initialized the elliptic curve as
> m0=13293998056584952174157235; E=ellinit([0,-432*m0]);
> 
> I tried to use the rational points found in
> https://arxiv.org/abs/math/0403116
> where the curve is apparently defined as
> E=ellinit([0,1,0,0,44182596082121121317135170025680399046545625711306]);
> and its independent points as
 Pvec=[[-30156002278649820, 4093799681127459731025817],[11364087102067560,
 6756491872572362690626342],[-20835788771691894,
 5927660006237675713476241],[1134264920569989390,
 1208031685828825118221478017],[8907565209691176834,
 26585114133655761890666064910],[111849199886121334,
 37992674604901443769570910],[11724873521668020,
 6767159346634715672034457],[-138658831412368575/4,
 12719819443574268333325811/8],[165971060901522240,
 67941788876402816577138982],[994768217796990,
 6647073075327662243966017],[532896351059436225/16,
 576457310785324883248677823/64]]
> but I can't replicate the result
> max{h_x(P_i) | 1 ≤ i ≤ 11} = 76.61
> and so must be doing something (several things?) wrong.

You should use a variant of my little game!
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)
%17 = [0,0,1,0,44182596082121121317135170025680399046545625711306]~
So you see, the curve equation is not quite right.

Cheers,
BIll.