The **regulator** of an elliptic curve $E$ defined over a number field $K$, denoted $\operatorname{Reg}(E/K)$, is the volume of $E(K)/E(K)_{tor}$ with respect to the **height pairing** $\langle -,-\rangle$ associated to the canonical height $\hat{h}$, i.e. $\langle P,Q\rangle = \frac{1}{2}(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q))$.

If the Mordell-Weil group $E(K)$ has rank $r$ and $P_1, \ldots, P_r \in E(K)$ generate $E(K)/E(K)_{\mathrm{tor}}$, then \[ \operatorname{Reg}(E/K) = \left|\det (\langle P_i, P_j \rangle )_{1\leq i,j \leq r}\right|, \] which is independent of the choice of generators.

Special cases are when $E(K)$ has rank $0$, in which case $E(K)/E(K)_{\mathrm{tor}}=0$ and $\operatorname{Reg}(E/K)=1$, and when $E(K)$ has rank $1$, in which case $\operatorname{Reg}(E/K)$ is equal to the canonical height $\hat{h}(P)$ of a generator $P$.

The canonical height used to define the regulator is usually *normalised* so that it is invariant under base change. Note that the regulator which appears in the Birch Swinnerton-Dyer conjecture is with respect to the non-normalised height.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-10-13 16:46:35

**Referred to by:**

- ag.regulator
- ec.bsdconjecture
- ec.canonical_height
- ec.q.bsd_invariants
- ec.q.bsdconjecture
- ec.q.canonical_height
- lmfdb/ecnf/main.py (line 789)
- lmfdb/ecnf/templates/ecnf-curve.html (lines 271-273)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 245)
- lmfdb/elliptic_curves/templates/ec-curve.html (line 316)

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

- 2020-10-13 16:46:35 by Andrew Sutherland (Reviewed)
- 2020-10-12 16:27:18 by Kiran S. Kedlaya
- 2020-10-07 11:46:46 by John Cremona
- 2019-02-08 07:41:06 by John Cremona (Reviewed)

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