Re: Question: trying to locate other Diophantine triples from certain elliptic curves

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

On Tue, Jul 09, 2024 at 09:15:38AM -0700, American Citizen wrote:
> This is only the first part of my two-part question, the other was working
> with the isogenous curves in short Weierstrass [0,0,0,A,B] format to see if
> they can be transformed to the E_triple(a,b,c) format with [a,b,c] a
> Diophantine triple.

You need your isogenous curve to have full two-torsion.
If it is the case, the same trick apply.
Isogenies of odd degree will preserve the two-torsion,
isogenies of even degree usually do not.

In your example, none of the isogenous curve has full two-torsion.

? [L,M]=ellisomat(E,,1);
? apply(e->elltors(ellinit(e))[2],L)
%9 = [[2,2],[2],[2],[2]]

so you are stuck with E.

Cheers,
Bill