Solution of certain Diophantine equations in Gaussian integers

In this article, we show that the quartic Diophantine equations $x^4\pm pq{y}^4=\pm z^2$ and $x^4\pm pq{y}^4=\pm i{z}^2$ have only trivial solutions for some primes $p$ and $q$ satisfying conditions $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}8),q\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}8)\text{ and }{\displaystyle \text{legendre}pq=-1}$. Here we have found the torsion of the two families of elliptic curves to find the solutions of given Diophantine equations. Moreover, we also calculate the rank of these two families of elliptic curves over the Gaussian field $\mathbb{Q}(i)$.