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

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

Hello:

I am working with elliptic curves associated with a Diophantine triple 
[a,b,c] such that a*b+1, a*c+1, and b*c+1 are all rational squares:

The equation of this curve is

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

in Weierstrass format.

For a selected triple [a,b,c] = t = [5/4, 5/36, 32/9], the curve becomes

(2)  E(5/4,5/36,32/9) = E(t) = [0, 6625/1296, 0, 2225/729, 2500/6561]

The minimal model of this curve is

(3)  M(t) =  [1, 0, 0, -593988, 172568592] with conductor 510450

GP-Pari ellisomat(E_triple) command gives four isogenous classes for (2) 
E(t) in [0,0,0,a,b] Weierstrass format

K_1 = [-28511425/5038848, 149142030625/29386561536]
K_2 = [-453599425/5038848, 9660698574625/29386561536]
K_3 = [3360575/5038848, 464053326625/29386561536]
K_4 = [-3935425/314928, -4396529375/459165024]

Using the expression for the c4 invariant to (1) above I came up with

(4)  F.c4 = (16*b^2 - 16*c*b + 16*c^2)*a^2 + (-16*c*b^2 - 16*c^2*b)*a + 
16*c^2*b^2

And using brute force searching for b,c, by solving for (4) = E(t).c4 
invariant = 28511425/104976 and the fact that the "a" variable
is a quadratic in F.c4, I came up with a list of over 164 triples which 
have the same c4 value for the elliptic curve E(t).

My apparently mistaken idea was that I could find isomorphic curves by 
matching the c4 value. This apparently is naive and too simplistic.

Realizing next that apparently the j-invariant must be the same, I 
located 34 triples creating elliptic curves with identical c4 and j 
invariants, and derived 13 unique triples from those which satisfy

E(t).c4 = 28511425/104976 and E(t).j = 1483326465959023993/34578915987456

Those triples are:

   [-7/36, 80/63, 35/12]
   [10/9, -5/36, 41/12]
   [28/9, 35/12, 415/252]
   [32/9, 83/36, 41/12]
   [40/27, -83/108, 205/108]
   [5/4, 10/9, -83/36]
   [5/4, 5/36, 32/9] (Diophantine triple)
   [7/36, -80/63, -35/12]
   [7/36, 28/9, 41/28]
   [8/3, 5/12, 205/108]
   [80/63, 41/28, -415/252]
   [9/4, -5/12, 40/27]
   [9/4, 8/3, 83/108]

Alas! only 1 triple is a Diophantine triple, the original t = [5/4, 
5/36, 32/9] My hope was that more Diophantine triples might appear. This 
list of triples seems infinite, btw.

All this work was only concerned with the first isogenous class (K_1) above.

I have a 2 part goal here.

1. Find all isogenous classes for E(t) (which is simple using the 
ellisomat() command) but obtaining
   elliptic curves in the format of (1), not the classical [0,0,0,a,b] 
format which the ellisomat command derives.

2. For each curve using a,b,c, quickly find all triples which give a 
matching E.j invariant value, and hopefully find other Diophantine triples.

The key idea here is that if we find E(t) from a Diophantine triple, 
then the other isogenous classes should likely also contain a 
Diophantine triple.

I did notice that the 4 isomorphism clases for E(t) do have different 
j-invariant values, so I am a bit confused at the moment on how we know that
all 4 are in this group.

Is there a way to quickly find triples for (1) given E(t) ? and, of 
course, for finding other triples for the other isogenous curves?

Thank you for a reply. I do appreciate any response, as this is the only 
way I have of discussing such things concerning elliptic curves with other
mathematicians, as I am retired.

Randall
Sat 6-July-2024