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

[American Citizen <website.reader3@gmail.com>]

John:

Yes, that is correct and with [a,b,c] being a Diophantine triple.

I can derive 20 elliptic curves from a given Diophantine sextuples, and the ellisomat() command finds (usually) 4 curves in the group, thus giving 80 curves in the short Weierstrass format [0,0,0,A,B]. But I need to find the isogenies so that I can find the E_triple(a,b,c) format and a,b,c a Diophantine triple. My conjecture is that we already have 20 curves which are derived from a triple, and 3 points can be found on that curve, which are a triple, it should be possible to find a similar curve and points in the isogeny group.

Randall

On 7/9/24 07:18, John Cremona wrote:

I think that Randall's question is this:  given an elliptic curve in either long [a1,a2,a3,a4,a6] or short [0,0,0,A,B] format, to determine whether there is an isomorphic curve of the form E_triple(a,b,c)  = [0,(a*b+a*c+b*c),0,(a*b*c)*(a+b+c),(a*b*c)2], and if so give the values of a,b,c and the transformation taking the input curve to the new curve.

Is that right?

John

On Tue, 9 Jul 2024 at 00:25, American Citizen <website.reader3@gmail.com> wrote:

To all:

I want to keep pursuing this question

We have an associated Elliptic Curve associated with a Diophantine Triple [a,b,c] where a*b+1, a*c+1 and b*c+1 are all squares.

(1)  E_triple(a,b,c)  = [0,(a*b+a*c+b*c),0,(a*b*c)*(a+b+c),(a*b*c)^{^2}]

When I do run the GP Pari command for a specific set curve, say using the triple [5/4, 5/36, 32/9], I get the Curve as [0, 6625/1296, 0, 2225/729, 2500/6561]

There apparently are 4 isogenous curves associated with this curve in Weierstrass [0,0,0,a,b] format:

[-28511425/5038848, 149142030625/29386561536]

[-453599425/5038848, 9660698574625/29386561536]

[3360575/5038848, 464053326625/29386561536]

[-3935425/314928, -4396529375/459165024]

This is all mathematically fine and correct, but what I am looking for is the urst transformations such that I get a set of 4 curves, with [a_i, b_i, and c_i] a Diophantine triple.

Can we find the urst transformations to move these [0,0,0,a,b] curves to the (1) format  and have [a,b,c] be a Diophantine triple?

Randall