Two infinite families of elliptic curves with rank greater than one

Abstract

We prove, using elementary methods, that each member of the infinite families of elliptic curves given by $E_m \colon y^2=x^3 - x + m^6$ and $E_m' \colon y^2=x^3 + x - m^6$ have rank at least $2$ and 3, respectively, under mild restrictions on $m$. We also prove stronger results for $E_m$ and $E_m'$ using more technical machinery.