# MathSciDoc: An Archive for Mathematician ∫

#### Number Theorymathscidoc:2204.24004

Documenta Math, 25, 2115-2147, 2020.11
We study elliptic curves of the form x^3+y^3=2^p 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(\sqrt[3]{p}) to obtain some examples of elliptic curves with rank one and non-trivial 2-part of the Tate-Shafarevich group.
@inproceedings{yukako2020a,
title={A Classical Family of Elliptic Curves having Rank One and the 2-Primary Part of their Tate-Shafarevich Group Non-Trivial},
author={Yukako Kezuka, and Yongxiong Li},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220422144311963603104},
booktitle={Documenta Math},
volume={25},
pages={2115-2147},
year={2020},
}

Yukako Kezuka, and Yongxiong Li. A Classical Family of Elliptic Curves having Rank One and the 2-Primary Part of their Tate-Shafarevich Group Non-Trivial. 2020. Vol. 25. In Documenta Math. pp.2115-2147. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220422144311963603104.