question on plane cubic curves <-> Weierstrass elliptic curve maps

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

Hello:

My goal here is to see if we can find maps with smaller powers of 
coefficients for forward/inverse maps between certain plane cubic curves 
and Weierstrass elliptic curves apparently matching the Jacobian of the 
cubic curves.

Let a plane body cubic equation in u,v be

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

where a,b are positive integers and u,v rational (usually > 1)

GP Pari ellfromeqn((1)) gives

(2) E_body(a,b) = [0, (a^2 + b^2)^2, 0, (2*a*b*(a-b)*(a+b))^2, 0]

provided that you substitute [x,y] for [u,v] in (1).

There is a rather complicated forward map from (1) -> (2) which I won't 
give. The inverse map has to be found also.

Magma returns a Weierstrass equation with higher powers of a,b as

(3) Magma_body(a,b) = [ 0, (-32*a^8*b^4 + 128*a^6*b^6 - 
32*a^4*b^8)/(a^12 - 14*a^10*b^2 + 63*a^8*b^4 - 100*a^6*b^6 + 63*a^4*b^8 
- 14*a^2*b^10 + b^12), 0, 256*a^8*b^8/(a^16 - 20*a^14*b^2 + 148*a^12*b^4 
- 492*a^10*b^6 + 726*a^8*b^8 - 492*a^6*b^10 + 148*a^4*b^12 - 20*a^2*b^14 
+ b^16), 0 ]

Concerning the forward/inverse maps, magma finds

Forward map (1) --> (3)

x = -8*a^3*b^3/(a^6 - 7*a^4*b^2 + 7*a^2*b^4 - b^6)*u;
y = -64*a^6*b^6/(a^12 - 14*a^10*b^2 + 63*a^8*b^4 - 100*a^6*b^6 + 
63*a^4*b^8 - 14*a^2*b^10 + b^12)*w;
z = (-1/2*a^2 + 1/2*b^2)/(a*b)*u + v;

in projective coordinates [x,y,z].

Inverse map (3) --> (1)

u=8*a^3*b^3/(a^6 - 7*a^4*b^2 + 7*a^2*b^4 - b^6)*x;
v=4*a^2*b^2/(a^4 - 6*a^2*b^2 + b^4)*x - 64*a^6*b^6/(a^12 - 14*a^10*b^2 + 
63*a^8*b^4 - 100*a^6*b^6 + 63*a^4*b^8 - 14*a^2*b^10 + b^12)*z;
w=y;

[u,v,w] projective coordinates.

I particularly like the fact that GP Pari is finding a much smaller 
Weierstrass format curve in smaller powers of a,b then Magma is giving. 
When it comes to using mwrank and other elliptic curve point finding 
programs, the curves with the smaller coefficients generally work much 
better, and it's not too hard to see that using (3) climbs up by using 
16th powers. I'd rather keep this at 4th powers not 16th powers.

My question is this, can we find simpler maps for the (1) -> (2) and (2) 
-> (1) mappings?

- Randall