High rank elliptic curves with torsion group $\mathbb {Z}/(2\mathbb {Z})$

Abstract

We develop an algorithm for bounding the rank of elliptic curves in the family $y^2=x^3-B x$, all of them with torsion group $\mathbb {Z} /(2 \mathbb {Z})$ and modular invariant $j=1728$. We use it to look for curves of high rank in this family and present four such curves of rank $13$ and $22$ of rank $12$.