The **rank** of an elliptic curve $E$ defined over a number field $K$ is the rank of its Mordell-Weil group $E(K)$.

The Mordell-Weil Theorem says that $E(K)$ is a finitely-generated abelian group, hence
\[ E(K) \cong E(K)_{\rm tor} \times \Z^r\]
where $E(K)_{\rm tor}$ is the finite torsion subgroup of $E(K)$, and $r\geq 0$ is the **rank**.

Rank is an isogeny invariant: all curves in an isogeny class have the same rank.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Vishal Arul on 2019-09-20 16:29:03

**Referred to by:**

- dq.ec.reliability
- ec.q.220110.bn1.bottom
- ec.q.234446.a1.bottom
- ec.q.37.a1.top
- ec.q.49.a2.bottom
- ec.q.65.a1.bottom
- ec.q.analytic_sha_value
- ec.q.bsd_invariants
- ec.q.bsdconjecture
- rcs.rigor.lfunction.ec
- lmfdb/ecnf/templates/ecnf-isoclass.html (line 68)
- lmfdb/elliptic_curves/elliptic_curve.py (line 642)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 181)
- lmfdb/elliptic_curves/templates/ec-index.html (line 42)
- lmfdb/elliptic_curves/templates/ec-isoclass.html (line 53)
- lmfdb/elliptic_curves/templates/ec-isoclass.html (line 59)
- lmfdb/elliptic_curves/templates/ec-search-results.html (line 32)
- lmfdb/elliptic_curves/templates/ec-stats.html (line 8)

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

- 2019-09-20 16:29:03 by Vishal Arul (Reviewed)
- 2019-02-08 10:04:19 by John Cremona (Reviewed)

**Differences**(show/hide)