Mordell-Weil generators (reviewed)

The Mordell-Weil group $E(K)$ of an elliptic curve $E$ over a number field $K$ is a finitely generated abelian group, explicitly described by giving a $\Z$-basis for the group, equivalently, a (minimal) set of Mordell-Weil generators, each of which is a rational point on the curve.

The generators consist of $r$ non-torsion generators, where $r$ is the rank of $E(K)$, and up to two torsion generators, which generate the torsion subgroup $E(K)_{\textrm{tor}}$.