Isomorphism between $\mathbb{P}^{1}$ and $V : X^{2} + Y^{2} = pZ^{2}$

My question is about an exercise from “arithmetic of elliptic curves”:

Let $ $ V : X^{2} + Y^{2} = pZ^{2}$ $ be a projective vareity in $ \mathbb{P}^2$ and $ p$ be a prime number.

prove that $ V$ is isomorphic to $ \mathbb{P}^{1}$ iff $ p \equiv 1$ (mod $ 4$ ).

and for $ p \equiv 3$ (mod $ 4$ ) no two of such vareities are isomorphic.


I was trying to solve this exercise but I have stuck in defining the morphism. And as it is the case that $ p \equiv 1$ (mod $ 4$ ) I think we should use an integer solution of $ a^{2} + b^{2} =p$ . but most of my efforts didn’t turn out to be a morphism. So if anyone could help with this, it would be great.