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

The torsion subgroup $E(K)_{\text{tors}}$ 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.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by David Farmer on 2019-05-07 05:55:46

**Referred to by:**

- ec.mordell_weil_group
- ec.q.bsd_invariants
- ec.q.bsdconjecture
- lmfdb/ecnf/templates/ecnf-index.html (line 136)
- lmfdb/ecnf/templates/ecnf-search-results.html (line 90)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 232)
- lmfdb/elliptic_curves/templates/ec-index.html (line 48)
- lmfdb/elliptic_curves/templates/ec-index.html (line 122)
- lmfdb/elliptic_curves/templates/ec-isoclass.html (line 20)
- lmfdb/elliptic_curves/templates/ec-search-results.html (line 19)
- lmfdb/elliptic_curves/templates/ec-search-results.html (line 127)

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

- 2019-05-07 05:55:46 by David Farmer (Reviewed)
- 2019-02-08 11:33:05 by John Cremona (Reviewed)

**Differences**(show/hide)