Is there a classification of smooth projective curves of genus $0$ over $\mathbb{Q}$?
I know that if the curve has a rational point, then it is isomorphic to $\mathbb{P}^1$.
The curve must embed as a degree $2$ curve in $\mathbb{P}^2$, so it has a point over some quadratic extension of $\mathbb{Q}$. This means the curve is a quadratic twist of $\mathbb{P}^1$.