Let $L$ be the Hilbert Class Field of $K=\mathbb{Q}(\sqrt{-d})$.
I already know, via Artin reciprocity, that $Gal(L/K) \cong CL(K)$. Another theorem (Cox 9.30) says that: $Gal(L/\mathbb{Q}) \cong CL(K) \rtimes Z_2$.
This gives me some basic information on the structure of $L$. Do there exist other theorems which give me even more structural information about $L$?
Next semester I will study complex multiplication and modular functions in the book of David Cox. The above question is limited to algebraic approaches/tricks.