A
C
M
CM
-field
K
K
defines a triple
(
G
,
H
,
ρ
)
(G,H,\rho )
, where
G
G
is the Galois group of the Galois closure of
K
K
,
H
H
is the subgroup of
G
G
fixing
K
K
, and
ρ
∈
G
\rho \in G
is induced by complex conjugation. A "
ρ
\rho
-structure" identifies
C
M
CM
-fields when their triples are identified under the action of the group of automorphisms of
G
G
. A classification of the
ρ
\rho
-structures is given, and a general formula for the degree of the reflex field is obtained. Complete lists of
ρ
\rho
-structues and reflex fields are provided for
[
K
:
Q
]
=
2
n
[K:\mathbb {Q}] = 2n
, with
n
=
3
,
4
,
5
n = 3,4,5
and
7
7
. In addition, simple degenerate Abelian varieties of
C
M
CM
-type are constructed in every composite dimension. The collection of reflex fields is also determined for the dihedral group
G
=
D
2
n
G = {D_{2n}}
, with
n
n
odd and
H
H
of order
2
2
, and a relative class number formula is found.