Hi Keerthi, I'm confused about one point. If $Stab(f)=H\leq G_{\mathbf{Q}}$, then you are showing that for all complex conjugation $c\in G_{\mathbf{Q}}$, $cHc^{-1}=H$. But this does not seem to imply that $L:=\bar{\mathbf{Q}}^{H}$ is a CM field. For example, if $L$ is Galois over $\mathbf{Q}$ and not CM, then the condition $cHc^{-1}=H$ is trivially satisfied since $H$ is normal in $G_{\mathbf{Q}}$. Am I right?

@Anton, I have 2 simple questions: what is the degree of your map $f$ (and how to compute it) and how do you compute a fiber $f^{-1}(y)$?

Yes you are right, I had in mind something different.