A quartic CM field field K is represented by invariants [D,A,B], where K = Q[x]/(x⁴+Ax²+B), and D is the discriminant of the totally real quadratic subfield (hence A²-4B = m²D for some m).

K	Quartic invariants	Cl(O_K)	Igusa invariants	K^r	Reflex invariants	Cl(O_K^r)	Igusa invariants
1)	[5, 349, 26669]	C₃ x C₃	■	23)	[26669, 193, 2645]	C₃ x C₃	■
2)	[5, 229, 11909]	C₃ x C₃	■	21)	[11909, 133, 1445]	C₃ x C₃	■
3)	[5, 274, 18749]	C₃ x C₃	■	22)	[18749, 137, 5]	C₃ x C₃	■
4)	[8, 326, 20297]	C₃ x C₃	■	---)	[20297, 163, 1568]	C₃ x C₃ x C₃	■
5)	[13, 233, 8629]	C₃ x C₃	■	19)	[8629, 281, 325]	C₃ x C₃	■
6)	[17, 190, 8753]	C₃ x C₃	■	20)	[8753, 95, 68]	C₃ x C₃	■
7)	[89, 383, 6212]	C₃ x C₃	■	13)	[1553, 211, 1424]	C₃ x C₃	■
8)	[97, 731, 122896]	C₃ x C₃	■	18)	[7681, 611, 76048]	C₃ x C₃	■
9)	[113, 343, 14468]	C₃ x C₃	■	17)	[3617, 667, 1808]	C₃ x C₃	■
10)	[193, 239, 10372]	C₃ x C₃	■	15)	[2593, 478, 15633]	C₃ x C₃	■
11)	[733, 154, 2997]	C₃ x C₃	■	---)	[37, 77, 733]	C₃	■
12)	[761, 43, 272]	C₃ x C₃	■	---)	[17, 86, 761]	C₃	■
13)	[1553, 211, 1424]	C₃ x C₃	■	7)	[89, 383, 6212]	C₃ x C₃	■
14)	[1901, 49, 125]	C₃ x C₃	■	---)	[5, 97, 1901]	C₃	■
15)	[2593, 478, 15633]	C₃ x C₃	■	10)	[193, 239, 10372]	C₃ x C₃	■
16)	[2857, 57, 98]	C₃ x C₃	■	---)	[8, 114, 2857]	C₃	■
17)	[3617, 667, 1808]	C₃ x C₃	■	9)	[113, 343, 14468]	C₃ x C₃	■
18)	[7681, 611, 76048]	C₃ x C₃	■	8)	[97, 731, 122896]	C₃ x C₃	■
19)	[8629, 281, 325]	C₃ x C₃	■	5)	[13, 233, 8629]	C₃ x C₃	■
20)	[8753, 95, 68]	C₃ x C₃	■	6)	[17, 190, 8753]	C₃ x C₃	■
21)	[11909, 133, 1445]	C₃ x C₃	■	2)	[5, 229, 11909]	C₃ x C₃	■
22)	[18749, 137, 5]	C₃ x C₃	■	3)	[5, 274, 18749]	C₃ x C₃	■
23)	[26669, 193, 2645]	C₃ x C₃	■	1)	[5, 349, 26669]	C₃ x C₃	■

Class number: [Non-normal] [Cyclic]

Class group C3 x C3 non-normal (D4) quartic CM field invariants: 23 fields

Class group C₃ x C₃ non-normal (D₄) quartic CM field invariants: 23 fields