I'm trying to find the set of solutions to a specific diophantine equation over $\mathbb{Z}[i]$. The equation is the following:
$ z_1^2 + z_2^2 + z_1*z_2 + 39 = 0$
with $ z_1$ (resp $z_2$) such that $\exists a,b \in \mathbb{Z} , z_1$ (resp $z_2$) $= a + ib $
Choosing $z_1 = a+ib$ and $z_2 = a-ib$, I can obtain a diophantine equation over $\mathbb{Z}$ and find a family of solutions, but I can't manage to describe other families. Are there specific techniques for this kind of equation ?
Regards