Torsion subgroup of an elliptic curve over $\mathbb Q$ (reviewed)

If $E$ is an elliptic curve defined over $\mathbb{Q}$, its torsion subgroup is the subgroup of the Mordell-Weil group $E(\mathbb{Q})$ consisting of all the rational points of finite order. It is a finite abelian group of order at most $16$ (by a theorem of Mazur), which is a product of at most $2$ cyclic factors. The "torsion structure" is the list of invariants of the group:

• $[]$ for the trivial group;
• $[n]$ for a cyclic group of order $n$ (only $n=2,3,4,5,6,7,8,9,10$ or $12$ occur for elliptic curves over $\mathbb{Q}$);
• $[n_1,n_2]$ with $n_1\mid n_2$ for a product of cyclic groups of orders $n_1$ and $n_2$ (only $[2,2m]$ for $m=2,4,6$ or $8$ occur over $\mathbb{Q}$).