Please note that although there is a very similarly titled question Exercise 1.10 from Silverman "The Arithmetic of Elliptic Curves" this question received no answers.
Let $p$ be an odd prime and $V_p\subseteq \mathbb P^2$ the variety $$V_p : X^2 + Y^2 = p Z^2 $$ I'm trying to show that $V_p \cong \mathbb P^1$ over $\mathbb Q$ if and only if $p\equiv 1\pmod 4$. The forward direction is easy: suppose $p\cong 3\pmod 4$. If $V_p$ is isomorphic to $\mathbb P^1$ over $\mathbb Q$ then $V_p (\mathbb Q) \cong \mathbb P^1 (\mathbb Q)$. But $X^2 + Y^2 = p Z^2$ has no solutions in integers and thus $V_p (\mathbb Q)=\varnothing$, and we know $\mathbb P^1 (\mathbb Q)\neq\varnothing$.
However I'm struggling to do the other direction because I can't cook up a good isomorphism $V_p \cong \mathbb P^1$ defined over $\mathbb Q$ when $p\equiv 1\pmod 4$. Can anyone help me out with this? Thank you!