Types of elliptic curves 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!
 A: 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. 
A: 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.
A: There are effective, albeit large, upper bounds for the integer solutions to any non-singular cubic
$$ ax^3+by^3+c+dx^2y+exy^2+fx^2+gx+hy^2+iy+jxy=0. $$
Such bounds are proven using linear forms in logarithms. I don't know the current best results, but the original result is in
[1] A. Baker and J. Coates, Integer points on curves of genus 1, Proc. Cambridge Philos. Soc. 67 (1970), 595-602.
Let $H$ be an upper bound for the coefficients of the cubic. They prove that any integer solution $(x,y)$ satisfies
$$ \max\{|x|,|y|\} \le \exp\exp\exp\left((2H)^{10^{3^{10}}}\right). $$
On the other hand, there are much better bounds known for equations of the form
$$ y^m = f(x), $$
which are sometimes call superelliptic equations (a term due to Lang, I believe). Anyway, here's one reference to get you started on practical methods for solving superelliptic euqations:
[2] Yuri F. Bilu, Guillaume Hanrot, Solving superelliptic Diophantine equations by Baker's method, Compositio Mathematica 112 (1998), 273-312.
