Re: Game: find the curve

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

On Tue, Apr 16, 2024 at 03:55:26PM +0200, Denis Simon wrote:
> Hi Randall,
> 
> Call [x,y] the coordinates of your point.
> The question is to find a linear relation between |y^2,x*y,y,x^3,x^2,x,1].
> The function lindep does the job.

Indeed:
? [X,Y]=P;F=lindep([X*Y,-X^2,Y,-X,-1,-X^3,Y^2])~
%1 = [0,-105941,-105941,-112234954810,-23483312417238135,1,1]
? E=ellinit(F[1..5]/F[7]);
? ellisoncurve(E,P)
%3 = 1
? ellglobalred(E)
%4 = [11223495481,[1,35314,0,52971],4,[11,2;9631,2],[[2,-9,0,2],[2,-1,0,2]]]
? ellminimaltwist(E)
%5 = 105941
? Et=elltwist(E,105941);
? ellidentify(Et)
%11 = [["11a1",[0,-1,1,-10,-20],[]],[105941,0,0,594508555628570

So actually the curve is the twist of 11a1 a.k.a X_0(11) by D=105941 = 11*9631.
Its conductor is 11223495481 = 11^2 * 9631^2 which is a bit large for ellheegner.

Cheers,
Bill.