a hopeful request

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

To all:

I have been looking for this for a long time.

Suppose I have an elliptic curve in Weierstrass format [a1,a2,a3,a4,a6] 
and a two-covering or quartic a*x^4 + b*x^3 + c*x^2 + d*x + e and they 
share the same invariants, the I invariant and the J invariant.

Is it possible to develop the forward and inverse maps for points on 
either the elliptic curve mapping to the quartic two-cover, or the 
points on the two-cover mapping back to the elliptic curve?

I would like to obtain the forward and inverse maps for points on either 
the elliptic curve -> quartic or quartic -> elliptic curve

Thank you for letting me know if this is possible and how to set this up 
in GP-Pari

Randall