I have a cubic curve that looks like
$$ a_0 x^3 + a_1 x^2 y + a_2 xy^2 + a_3 y^3 = b $$
with $a_0, a_1, a_2, a_3$, and $b$ all integers, and $a_0$ and $b$ nonzero. I'm not sure but I think in my situation $a_3$ is nonzero as well.
I'd like to know whether this curve has infinitely-many integer points. As I understand it, Siegel's Theorem implies that the curve has only finitely-many integer points, so long as its genus is positive. However, algebraic geometry is pretty far from my normal area, and I have no idea how to determine the genus of such a curve. Hence,
Question: Does such a curve necessarily have positive genus? If not, is there an easily-described counterexample? Or, is there a proof that such a curve has only finitely-many integer solutions that does not go through Siegel's theorem?
A cubic curve is either rational (
genus $0$) if it has a double point or a cusp, or elliptic (genus 1) if it is a smooth curve.If a cubic is elliptic, its points can be given the structure of an abelian group, and Mordell-Weil's theorem states that the curve on $\mathbf Q$ is a finitely generated abelian group. To make the structure of group effective, you must use Weierstraß normal form.
Siegel's theorem says an elliptic cubic has a finite number of points with first coordinates in the ring of integers of an algebraic number field.