Reflex field (awaiting review)

Let $K$ be a CM number field and let $\overline{\mathbb{Q}}$ be the algebraic closure of $\Q$ in $\C$. A subset $\Phi \subset \mathrm{Hom}(K, \overline{\mathbb{Q}})$ is called a CM type if for every embedding $\iota \in \mathrm{Hom}(K, \overline{\mathbb{Q}})$ either $\iota \in \Phi$ or $\overline{\iota} \in \Phi$, but not both, where $\overline{\iota}$ is the complex conjugate of $\iota$.

Given a CM field $K$ and a CM type $\Phi$, the reflex field is the fixed field inside $\overline{\Q}$ corresponding to the subgroup $\{ \rho \in \Gal(\overline{\Q}/\Q) : \rho \Phi = \Phi \}$ of $\Gal(\overline{\Q}/\Q)$. A CM type $\Phi$ and its complement $\overline{\Phi}$, which is the same as the set of complex conjugate embeddings, have the same reflex field. The number of complex conjugate pairs of CM types is $2^{g-1}$, where $2g=[K:\Q]$, the degree of $K$ over $\Q$.

To specify a CM type $\Phi$ for the CM field $K=\Q(a)$:

1. fix an order $ (\iota_1,\overline{\iota_1}), \dots, (\iota_g,\overline{\iota_g}) $ of the pairs of complex embeddings of $K$;
2. then $\Phi=(\varphi_1,\dots,\varphi_g)$ where $\varphi_j\in\{\iota_j,\overline{\iota_j}\}$ for $1\le j\le g$;
3. now $\Phi$ can be encoded by the list $(\text{sign}(\text{im}(\varphi_1(a))),\dots,\text{sign}(\text{im}(\varphi_g(a))))$.

The CM types in the LMFDB are grouped in Galois orbits under the action of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ described above.