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 $\mathbb{Q}$).

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- Last edited by John Jones on 2018-06-19 15:21:13

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- 2018-06-19 15:21:13 by John Jones (Reviewed)