This is a problem about an elliptic quartic in

is defined by

and

Let be the point

Let be the projection from to the plane .

I need to show that induces an isomorphism of with the plane cubic curve defined by:

minus the point

The point of the exercise is to prove that is irreducible and nonsingular. This follows after showing that is an isomorphism since then we can show that is irreducible and nonsingular which is straightforward.

I have been trying to show that is contained in and vice versa.

Am I correct to right the as the equations that define with ?

if so (excluding the point )

But the algebra is not working out for me. Is there another approach?