Question on elliptic curves with same conductor and Diophantine triples

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

Subject: Question on using GP-Pari to uncover rational polynomial 
relationships for certain elliptic curves.

Hello:

I am working with Diophantine triples, and the elliptic curves generated 
by them.

A Diophantine triple is typically [a,b,c] where a,b,c are rational, but 
the additional requirement is that any pair (a,b), (a,c), (b,c) 
satisfies the Diophantine requirement that a pair product be 1 less than 
a (rational) square.

(1)  a*b + 1 = s^2 ; a*c + 1 = t^2 ; b*c + 1 = u^2

Example:

  Take [1,3,8] as a Diophantine triple. Clearly 1*3 + 1 = 4 = 2^2 and 
1*8+1 = 9 = 3^2 and 3*8+1 = 25 = 5^2 so this is a Diophantine triple.

  An example of a rational Diophantine triple is [5/4, 5/36, 9/32] with 
their respective squares (13/12)^2,  (7/3)^2 and (11/9)^2

There is an elliptic curve associated with each Diophantine triple, by 
definition, the Weierstrass form is:

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

Such elliptic curves created by Diophantine triples usually have high 
rank, interestingly enough. For example the triple [755/567, 12376/405, 
5544/47045] generates a rank 9 curve.

I have discovered some interesting relationships among the triples and 
elliptic curves which have the same  conductor.

Take the following two pairs as examples:

Example 1

conductor                  elliptic curve Diophantine Triple
31141110   [0, -208619/396900, 0, -36542/33075, 7744/11025] [35/36, 
-27/35, 352/315]
31141110   [0, 868006/38025, 0, 1646161/38025, 559504/38025] [12/65, 
143/60, 340/39]

Example 2

conductor                         elliptic curve Diophantine Triple
116867520  [0, 278088/4844401, 0, -11768632283520/23468221048801, 
296284262400/23468221048801]  [40/9, 162/961, -756/5041]
116867520  [0, 952/9, 0, 6765440/2187, 
11907174400/531441]                                      [22/9, 40/9, 124/9]

Please notice that in the second example the two triples contain a 
common 40/9 tuple.

Two questions:

1. How can we use GP-Pari to discover the rational polynomial 
relationship between the two Diophantine triples which share the same 
(minimal) elliptic curve?

2. Given one triple [a,b,c] can we find another triple [d,e,f] 
associated with the same minimal elliptic curve? This process is 
essentially generating other elliptic curves with the same conductor but 
from other Diophantine triples.

Please bear in mind that the Diophantine relationship (1) has to 
satisfied in any triple we find.

To me is seems that we are working with different values for the a2, a4 
and a6 Weierstrass coefficients but such that they determine the same 
minimal curve and identical conductor.

Thank you for considering my questions.

Randall