Types of elliptic curves

Question

I'm trying to research elliptic curves, and I always get the generic equation $$y^2 = a_0 x^3 + a_1 x^2 + a_2 x + a_3.$$ However, I'm looking for information on an equation like $$y^3 = a_0 x^3 + a_1 x^2 + a_2 x + a_3$$ or an equation with cubes on both sides. I can't seem to find anything... are they called something else? Are there any papers I could read on them? Thanks!

Answer 1

Such equations are called Cubic plane curves; references are given here. The projective version is given by $F(x,y,z)=0$ where $F$ is a non-zero linear combination of the third-degree monomials $$ x^3, y^3, z^3, x^2y, x^2z, y^2x, y^2z, z^2x, z^2y, xyz. $$ For $z=1$ we obtain the affine version. Any non-singular cubic curve can be transformed into the Weierstrass equation of an elliptic curve.

Answer 2

Have a look at Silverman and Tate's "Rational Points on Elliptic Curves". There, in page 22, they tell you how to transform any non-singular cubic into a Weierstrass form. The reason why you don't see much work on curves of the form $y^3=x^3+\cdots$ is that we first bring it to a Weierstrass form and then work there.