Mazur's theorem

Definition
Let $$E$$ be an elliptic curve over the field $$\mathbb{Q}$$ of rationals. Then, the torsion subgroup of $$E$$ (viz the subset of the elliptic curve group that comprises points of finite order) is isomorphic, as an abstract Abelian group, to one of the following:


 * The cyclic group of order $$n$$, where 1 \le n \le 10</math? or $$n = 12$$
 * The direct product $$\Z/2\Z \oplus \Z/2n\Z$$ where $$1 \le n \le 4$$