# Smooth projective curves of genus 0

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 $$\mathbb{P}^1$$.

• Doesn't Cassels say in his book on elliptic curves: "Fact. A genus 0 curve is equivalent to a line or conic"? – Richard Martin Nov 13 '18 at 13:07
• Your classification is correct. Are you asking for a stronger classification in some sense? – Samir Canning Nov 13 '18 at 14:57
• I would like a list of isomorphism classes of curves. I know each curve can be embedded as a conic in $\mathbb{P}^2$, but it isn't obvious (to me, at least) when two conics are isomorphic. – User Nov 13 '18 at 16:30
• These kind of varieties are called Brauer-Severi varieties. The question is equivalent to asking for a classification of quadratic forms in three variables over $\mathbb{Q}$. I am not sure how useful is this for you. – random123 Nov 15 '18 at 8:01