Mazur’s theorem on torsion of elliptic curves

Theorem 1 (Mazur).

Let $E\mathrm{/}\mathrm{Q}$ be an elliptic curve. Then the torsion subgroup$E_{\mathrm{torsion}}\mathit{}\mathrm{(}\mathrm{Q}\mathrm{)}$ is exactly one of thefollowing groups:

$$\mathbb{Z}/N\mathbb{Z}\mathit{\hspace{1em}}1\le N\le 10\mathit{\hspace{1em}}or\mathit{\hspace{1em}}N=12$$

$$\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2N\mathbb{Z}\mathit{\hspace{1em}}1\le N\le 4$$

Note: see Nagell-Lutz theorem for an efficient algorithm to compute the torsion subgroup of an elliptic curve defined over $\mathbb{Q}$.