A Classical Family of Elliptic Curves having Rank One and the $2$-Primary Part of their Tate-Shafarevich Group Non-Trivial

Abstract

We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2 \mod 9$ or $p\equiv 5 \mod 9$. We first show that the $3$-part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their $2$-Selmer group to the $2$-rank of the ideal class group of $\mathbb{Q}(\sqrt[3]{p})$ to obtain some examples of elliptic curves with rank one and non-trivial $2$-part of the Tate-Shafarevich group.