I've been studying about the arithmetic of rational algebraic curves, i.e., curves defined by $C:f(x,y)=0$, where $f\in\mathbb{Q}[x,y]$ is irreducible, with $\deg(f)=d$.
So far, here is what I've found:
(i) for $d=1$ (line): it's easy to describe all rational points of $C$
(ii) for $d=2$ (conics): if there is one rational point, we can describe all others by using a parametrization.
(iii) for $d\geq 4$: by Faltings' theorem, there are finitely many rational points in $C$.
(assuming $C$ non-singular)
So my question is about $d=3$: why do people always talk about elliptic curves (which have the form $y^2=x^3+Ax+B$), and not about other curves like, for example, $y^3-y^2-x^3+2x=0$? What happens with cubics in general?
A non-singular cubic with at least one rational point can be transformed to an elliptic curve.
Singular irreducible cubics are rational curves, and can be parameterised by rational functions just as conic can.