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 03:18:37PM +0100, 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.

Indeed, but this seems a bit circular: the equation of E_triple is

y^2 = (x+c*b)*(x+c*a)*(x+b*a)

This is equivalent to look for curves with full 2-torsion,
y^2 = (x+A)*(x+B)*(x+C) with A*B*C=z^2 is a square.
by setting A = c*b, B = c*a, C = b*a, z=a*b*c
and conversely 
a=z/A b=z/B c= z/C

To preserve that, we need to pick a variable change of the form [u,r,0,0] with lead to 
y^2 = (x+(A+r)/u^2)*(x+(B+r)/u^2)*(x+(C+r)/u^2)
so we want (A+r)*(B+r)*(C+r) to be a square which is identical to asking
for a point on E_triple !

So if (x,y) is a point on E(a,b,c) then 
y/(c*b+x) , y/(c*a+x) and y/(b*a+r) is another triple.

An example:

E_triple(a,b,c)  = [0,(a*b+a*c+b*c),0,(a*b*c)*(a+b+c),(a*b*c)^2]
E=ellinit(E_triple(5/4,5/36,32/9));
ellrank(E)
F=ellchangecurve(E,[1, -235/54,0,0])
E_triple(475/36,-19/60,-50/171)==F[1..5]

Cheers,
Bill