$\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks

Somnath Jha, Dipramit Majumdar, Pratiksha Shingavekar

We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b{)}^2$ with $a,b\in \mathbb{Z}$. These elliptic curves have a rational $3$-isogeny, say $\phi$. We give an upper and a l…