comment on reverse map

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

After manually working about a day, I have an expression for the reverse 
map from an elliptic curve connected with body cuboids taking the point 
[x,y] to s, a Pythagorean generator ratio.

Let A=(a+b)*(a-b) and B=2*a*b where a,b are positive integers and a>b

Body Cuboids
Plane cubic: A/B = (u^2-1)/(2*u) * (2*v)/(v^2-1)  from John Leech, "The 
Rational Cuboid Revisited"

body_plane_cubic(u,v) = 2*a*b*v*(u+1)*(u-1) - (a-b)*(a+b)*u*(v+1)*(v-1) = 0

Elliptic_Curve_body(a,b) = [0, (a^2 + b^2)^2, 0, (2*a*b*(a-b)*(a+b))^2, 
0]  from ellfromeqn(body_plane_cubic(x,y))

r=11/2 (where a=11 and b=2) from the Body Cuboid Pythagorean generators 
Triad [11/2, 8/5, 6/5]
Curve: [0, 15625, 0, 26501904, 0]

Rank 1 Mordell-Weil basis: [-13068, 300564] Height: 2.007768259240
Curve is Z4xZ2 so there are 8 torsion points

Inverse map from point on elliptic curve to body cuboid ratio w
using a non-torsion point on the curve [x,y]

s = abs((y+((2*a*b^5 - 4*a^3*b^3 + 2*a^5*b) + 2*a*b*x))/(y-((2*a*b^5 - 
4*a^3*b^3 + 2*a^5*b) + 2*a*b*x)))
?(s > 1) : w=s ; w=1/s

I did notice that the forward map using X=8/5 and Y=6/5 gives the point 
[-8625969/3721, 22782250260/226981] but of height 8.0310730369 which 
seems to be 2x?

Randall