There are infinitely many elliptic curves over the rationals of rank 2

We show that there are infinitely many elliptic curves $E\mathrm{/}\mathbb{Q}$, up to isomorphism over $\bar{\mathbb{Q}}$, for which the finitely generated group $E(\mathbb{Q})$ has rank exactly $2$. Our elliptic curves are given by explicit models and their rank is shown to be $2$ via a $2$-descent. That there are infinitely many such elliptic curves makes use of a theorem of Tao and Ziegler.