Let $K\subset\mathbb{C}$ be a number field. There is a surjective map $\phi:\mathbb{P}^1(K)\to Cl(K)$ from the field to the class group, sending $[\alpha:\beta]$ to the class of the ideal $(\alpha,\beta)$. What does the fibre over an ideal class ``look like''?
I'm specifically interested in $K$ imaginary quadratic and whether or not there are circles in each of the fibres. For instance, $\mathbb{Q}=\mathbb{R}\cap K$ is contained in the fibre over the principal class.
Are there any elementary observations to make concerning $\phi$? Or references? Thanks.
Here are some pictures for reference, the two classes in $\mathbb{Q}(\sqrt{-5})$
http://math.colorado.edu/~rohi1040/randomshare/sqrt5princlass.pdf
http://math.colorado.edu/~rohi1040/randomshare/sqrt5otherclass.pdf
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As noted in one of the answers below, $SL_2(\mathcal{O}_K)$ acts on $\mathbb{P}^1(K)$ as fractional linear transformations and the fibres of $\phi$ are the $SL_2(\mathcal{O}_K)$-orbits. So, for instance, $\phi^{-1}(1)\cong SL_2(\mathcal{O}_K)/Stab(\infty)$. However, I don't think the fibres are isomorphic, and I don't know how to pick out the conjugacy classes of point-stabilizers (maximal elementary parabolic subgroups of $SL_2(\mathcal{O}_K)$?).
The fibers are precisely the orbits of the natural action of $SL_2(\mathcal{O}_K)$ on $\mathbb{P}^1(K)$. See this note by Keith Conrad for details.