Ogg's torsion conjecture for elliptic curves over the rationals

Let $E$) be an elliptic curve over $Q$. The rational torsion subgroup $E(Q)_\mathrm{tor}$ is one of the following fifteen groups:

$$\mathbb{Z}/N\mathbb{Z},\quad 1\leq N\leq 10\text{ or }N=12;$$ $$\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2N\mathbb{Z},\quad 1\leq N\leq 4.$$

Ogg's torsion conjecture. If $E$ is an elliptic curve over $\mathbb{Q}$, then its rational torsion subgroup has one of these forms; equivalently, $X_1(N)$ has no $\mathbb{Q}$-rational points besides cusps unless $1\leq N\leq 10$ or $N=12$. This is the classification of rational torsion subgroups of elliptic curves over $\mathbb{Q}$ and is equivalent to a statement about rational points on modular curves. It was proved by Mazur.