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.
- Review status: reviewed
- Last edited by John Cremona on 2019-02-08 12:14:19
- 2019-02-08 12:14:19 by John Cremona (Reviewed)