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

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.