Tricks to find the Hilbert Class field of a quadratic extension?

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.

• Like the genus field? – awllower Feb 15 '14 at 15:37
• @yannickvda Can you tell me the name of the book by Cox ? – learning_math Nov 20 '16 at 11:41

Artin's reciprocity law lets you compute the structure of the Galois group of the Hilbert class field over a base field: if $\sigma$ is an automorphism of $K/k$ and if $L$ is the Hilbert class field of $K$, and if you let $\tau = \big(\frac{L/K}{c}\big)$ denote the Artin symbol of an ideal class, then $\tau^{-1} \sigma \tau$ is the Artin symbol $\big(\frac{L/K}{c^\sigma}\big)$. There are a couple of details to be filled in, as you will see.