= Rank of an elliptic curve
{title2=$r=\dim_{\mathbb Q}(E(K)\otimes_{\mathbb Z}\mathbb Q)$}
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.
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