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

TL;DR

This paper proves the existence of infinitely many elliptic curves over the Gaussian rationals with a specific $j$-invariant and rank 2, using descent methods and a prime constellation theorem.

Contribution

It establishes the infinitude of rank 2 elliptic curves over $Q(i)$ with $j=1728$ not derived from $Q$, combining descent techniques and prime constellation results.

Findings

Infinitely many such elliptic curves exist over $Q(i)$.

All these curves have rank exactly 2.

They are not base changes from $Q$.

Abstract

We prove that there exist infinitely many elliptic curves over $\mathbb{Q}(i)$ with $j$-invariant $1728$ and rank exactly $2$ which are not obtained by base change from $\mathbb{Q}$. The rank of each such curve is determined via 2-isogeny descent, and the existence of infinitely many such curves follows from Tao's constellation theorem for Gaussian primes.