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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 $E(K)/E(K)_{tor}$, 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 $E(K)/E(K)_{tor}=0$ 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 $P$.

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  • Last edited by John Cremona on 2019-02-08 07:41:06
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