For an elliptic curve $E$ defined over $\R$ with period lattice $\Lambda$, the **real period** $\Omega$ is the least positive element of $\Lambda\cap\R$ multiplied by the number of components of $E(\R)$.

When an elliptic curve is defined by means of a Weierstrass equation, the period lattice $\Lambda$ is the lattice of periods of the invariant differential $dx/(2y+a_1x+a_3)$. Different Weierstrass models defining isomorphic curves have period lattices which are **homothetic**, meaning that they differ by a nonzero multiplicative constant. When we speak of **the** period lattice or **the** real period for an elliptic curve defined over $\Q$, we always mean the lattice and period associated with a minimal equation.

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- Last edited by John Cremona on 2019-02-08 12:14:19

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- 2019-02-08 12:14:19 by John Cremona (Reviewed)