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 $n=1,2,3,4$.

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2019-02-08 11:31:12

**Referred to by:**

- ec.mordell_weil_group
- ec.q.torsion_growth
- ec.rank
- ec.torsion_order
- lmfdb/ecnf/ecnf_stats.py (line 80)
- lmfdb/ecnf/main.py (line 450)
- lmfdb/ecnf/main.py (line 877)
- lmfdb/ecnf/templates/ecnf-curve.html (line 220)
- lmfdb/elliptic_curves/elliptic_curve.py (line 434)
- lmfdb/elliptic_curves/elliptic_curve.py (line 1177)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 133)
- lmfdb/elliptic_curves/templates/ec-isoclass.html (line 24)

**History:**(expand/hide all)

- 2019-02-08 11:31:12 by John Cremona (Reviewed)