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 ≡ 2 mod 9 or p ≡ 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 Q( 3√ p) to obtain some examples of elliptic curves with rank one and non-trivial 2-part of the Tate-Shafarevich group.