Rank of an elliptic curve ($r={\dim }_{\mathbb{Q}}(E(K){\otimes }_{\mathbb{Z}}\mathbb{Q})$)

For an elliptic curve over a number field $K$, the Mordell-Weil theorem writes $E(K)$ as a finite torsion subgroup plus a free abelian group of finite rank of an abelian group $r$. This integer is the rank of the elliptic curve over $K$. It can increase under field extension. A two-isogeny descent supplies rank bounds from rational square-class images; the canonical height pairing can certify independence of rational points.