## 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.