= Mordell-Weil saturation
{c}
{title2=$H^{\rm sat}=\{P\in E(K):nP\in H\text{ for some }n\ge1\}$}
The saturation of a subgroup $H$ of the <Mordell-Weil group> consists of all points with a positive multiple in $H$. Saturation includes the full <torsion subgroup>. For a full-rank $H$ containing that torsion, it equals $E(K)$, but the index of $H$ may exceed one. Checking that $\ell P\in H$ forces $P\in H$ for every <prime> divisor of a certified index bound proves that $H$ is already the whole group. A <canonical height pairing> proves independence of found points; saturation answers the separate question of whether a full-rank subgroup is the full group.
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