Infinitely many elliptic curves over $\mathbb{Q}(i)$ with rank 2 and $j$-invariant 1728 (2506.17605v1)

Abstract: We prove that there exist infinitely many elliptic curves over $\mathbb{Q}(i)$ of the form $y^2 = x^3 - \gamma^2 x$, where $\gamma \in \mathbb{Z}[i]$, with rank 2. In addition, we prove that if $p$ and $q$ are rational twin primes with $p \equiv 5 \bmod 8$, then $y^2 = x^3 + p q x$ has rank 2 over $\mathbb{Q}(i)$. Lastly, we prove that if $p$ is a rational prime of the form $p = a^2 + b^4$ (of which there are infinitely many) and $p \not\equiv 1 \bmod 16$, then $y^2 = x^3 - p x$ has rank 2 over $\mathbb{Q}(i)$.