Torsion subgroup of an elliptic curve (reviewed)

For an elliptic curve $E$ over a field $K,$ the torsion subgroup of $E$ over $K$ is the subgroup $E(K)_{\text{tor}}$ of the Mordell-Weil group $E(K)$ consisting of points of finite order. For a number field $K$ this is always a finite group, since by the Mordell-Weil Theorem $E(K)$ is finitely generated.

The torsion subgroup is always either cyclic or a product of two cyclic groups. The torsion structure is the list of invariants of the group:

• $[]$ for the trivial group;
• $[n]$ for a cyclic group of order $n>1$;
• $[n_1,n_2]$ with $n_1\mid n_2$ for a product of non-trivial cyclic groups of orders $n_1$ and $n_2$.

For $K=\Q$ the possible torsion structures are $[n]$ for $n\le10$ and $n=12$, and $[2,2n]$ for $n=1,2,3,4$.