Elliptic Curves of High Rank with Nontrivial Torsion Group over $\Q$

Abstract

We construct elliptic curves over $\Q$ of high Mordell--Weil rank, with a nontrivial torsion subgroup. We improve the rank records for the cases $\Z/2\Z \times \Z/2\Z$, $\Z/3\Z$, $\Z/4\Z$, $\Z/5\Z$, $\Z/6\Z$, $\Z/7\Z$ and $\Z/8\Z$.