The Selmer groups of elliptic Curves $E_{n}: y^2=x^3+nx$

Abstract

We give the computable conditions for the Selmer groups of elliptic curves $E_{n}$ and $E_{n}'$ related to the isogenies of degree 2, where $n$ is a square-free integer. In many cases, we can compute the Selmer groups by linear systems of equations defined over finite field $\mathbb{F}_{2}.$ As an application, we find many elliptic curves with the Mordell-Weil rank 0. Finally, we also obtain a sharp upper bound for the Mordell-Weil rank of $E_{n}$.