Geodesic
A Classical Family of Elliptic Curves having Rank One and the $2$-Primary Part of their Tate-Shafarevich Group Non-Trivial
Documenta mathematica, Tome 25 (2020), pp. 2115-2147.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 11G05, 11R29
Keywords: elliptic curves, complex multiplication, Tate-Shafarevich group, ideal class group 
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     title = {A {Classical} {Family} of {Elliptic} {Curves} having {Rank} {One} and the {\(2\)-Primary} {Part} of their {Tate-Shafarevich} {Group} {Non-Trivial}},
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Kezuka, Yukako; Li, Yongxiong. A Classical Family of Elliptic Curves having Rank One and the \(2\)-Primary Part of their Tate-Shafarevich Group Non-Trivial. Documenta mathematica, Tome 25 (2020), pp. 2115-2147. http://geodesic.mathdoc.fr/item/DOCMA_2020__25__a12/