Infinite family of elliptic curves of rank at least 4

Abstract

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