Infinite family of elliptic curves of rank at least 4

Abstract

We investigate $\mathbb{Q}$-ranks of the elliptic curve $E_t$: $y^2+txy=x^3+tx^2-x+1$ where $t$ is a rational parameter. We prove that for infinitely many values of $t$ the rank of $E_t(\mathbb{Q})$ is at least 4.