Torsion order of an elliptic curve (reviewed)

The torsion order of an elliptic curve $E$ over a field $K$ is the order of the torsion subgroup $E(K)_{\text{tor}}$ of its Mordell-Weil group E(K).

The torsion subgroup $E(K)_{\text{tor}}$ is the set of all points on $E$ with coordinates in $K$ having finite order in the group $E(K)$. When $K$ is a number field (for example, when $K=\Q$) it is a finite set, since by the Mordell-Weil Theorem, $E(K)$ is finitely generated.

When $K=\Q$ the torsion order $n$ satisfies $n\le16$, by a theorem of Mazur.