Infinitely Many Elliptic Curves of Exact Ranks Five, Six, and Seven
We prove that for each $r\in\{5,6,7\}$, there exist infinitely many elliptic curves over $\mathbb{Q}$, pairwise nonisomorphic over $\overline{\mathbb{Q}}$, whose Mordell–Weil rank over $\mathbb{Q}$ is exactly $r$. Zywina recently proved the analogous statement for $0\le r\le 4$ over every number field, and our proof follows his strategy. For each $r$, we give an explicit pair of $2$-isogenous elliptic curves whose discriminants are, up to constants, products of powers of twelve binary linear forms in two integer parameters. The families are obtained from Zywina's curves of rank four, or from…
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