Rank of an elliptic curve over a number field (reviewed)

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.