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 $\mathbb{P}^1$.

  • $\begingroup$ Doesn't Cassels say in his book on elliptic curves: "Fact. A genus 0 curve is equivalent to a line or conic"? $\endgroup$ – Richard Martin Nov 13 '18 at 13:07
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    $\begingroup$ Your classification is correct. Are you asking for a stronger classification in some sense? $\endgroup$ – Samir Canning Nov 13 '18 at 14:57
  • $\begingroup$ 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. $\endgroup$ – User Nov 13 '18 at 16:30
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    $\begingroup$ 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. $\endgroup$ – random123 Nov 15 '18 at 8:01

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