The Mordell-Weil group $E(\mathbb{Q})$ of an elliptic curve is a finitely generated Abelian group, that is, of the form $\mathbb{Z}^r \oplus T$, where $r \geq 0$ is the rank of $E(\mathbb{Q})$, and $T$ a finite torsion subgroup. This table contains the highest known ranks of an elliptic curves/$\mathbb{Q}$ with given torsion subgroup $T$, for all possible $T$.
Parameters
$T$
— torsion subgroup of $E(\mathbb{Q})$
Comments
(1)
For the long list of references leading to this table, see [3].
References
[1]
Noam D. Elkies, "Three lectures on elliptic surfaces and curves of high rank" (arXiv)
[2]
Noam D. Elkies, Zev Klagsbrun, "New Rank Records For Elliptic Curves Having Rational Torsion" (arXiv)