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Bill Allombert on Fri, 03 May 2024 14:05:36 +0200
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Re: h_x of points on a rank-11 elliptic curve
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- To: pari-users@pari.math.u-bordeaux.fr
- Subject: Re: h_x of points on a rank-11 elliptic curve
- From: Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>
- Date: Fri, 3 May 2024 14:05:23 +0200
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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.