Highest known ranks of elliptic curves over $\mathbb{Q}$ with prescribed torsion subgroup
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Numbers
$T$ 
$r$
$0$:
28
$\mathbb{Z}/2\mathbb{Z}$:
20
$\mathbb{Z}/3\mathbb{Z}$:
15
$\mathbb{Z}/4\mathbb{Z}$:
13
$\mathbb{Z}/5\mathbb{Z}$:
9
$\mathbb{Z}/6\mathbb{Z}$:
9
$\mathbb{Z}/7\mathbb{Z}$:
6
$\mathbb{Z}/8\mathbb{Z}$:
6
$\mathbb{Z}/9\mathbb{Z}$:
4
$\mathbb{Z}/10\mathbb{Z}$:
4
$\mathbb{Z}/12\mathbb{Z}$:
4
$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$:
15
$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$:
9
$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$:
6
$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$:
3
Definition
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)
Links
Data properties
Numbers are of type: integer
Table is complete: yes
Sources of data: For the long list of references leading to this table, see [3].
Reliability: It is unknown whether examples of higher rank exist.