Suppose $G$ is an abelian group of a cubic over $\mathbb{C}$. I need to find group order of $G_{n}=\lbrace x \in G| \exists n \in \mathbb{N}: nx=0 \rbrace$. I solved the problem for $n=2$: $\mathbb{Z}_{2}\oplus \mathbb{Z}_{2}$ (consists of points in which tangent lines are vertical); and for $n=3$: $\mathbb{Z}_{3}\oplus \mathbb{Z}_{3}$ (consists of inflection points). So, it seems that the answer is $n^{2}$...
Thank you in advance!
The $\Bbb C$ points of a complex elliptic curve form a torus: see here, for instance. So the $n$-torsion subgroup is exactly $(\Bbb Z/n)^2$.