# All complex quadratic fields K with discriminant -100000 < d < 0 and 3-class group (3,3)

First, we give a coarse classification by means of the 3-class numbers of the four associated complex cubic fields (L1,L2,L3,L4).
Most interesting are the cases with one or more occurrences of 3-class number 9 or 27.

(Computed by Fung and Williams in 1990 at the University of Manitoba, Winnipeg City [3,4].
Class field theoretic and group theoretic interpretation in May 2006 by our most recent sophisticated theorems [5].)

It should be pointed out that Brink has determined the capitulation type
of all fields with discriminant marked by an asterisk (*)
in his 1984 thesis [2],
based on class group computations of Wada [1]
for odd discriminants d > -24000 and even discriminants d > -96000.

Our own table of capitulation types for all discriminants d > -50000
has been constructed in 2003 without the knowledge of Brink's thesis,
which came to our disposal with delay in 2006.
It is contained in [6] and confirms Brink's results with a single exception d=-49128 (type D.10 instead of D.5).

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e = 3:
3.0:(3,3,3,3)...D,G.19,H0 (Ground State)
3.1:(9,3,3,3)...E1,G.16,H1 (First Excited State)
3.2:(27,3,3,3)...E2 (Second Excited State)

e = 5:
5.0:(9,9,3,3)...F0,GV,HV (Ground State of Variant)
5.1:(27,9,3,3)...F1 (First Excited State of Variant)

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1. 3896*: 3.0
2. 4027*: 3.0
3. 6583*: 3.0
4. 8751*: 3.0
5. 9748*: 3.1
6. 12067*: 3.0
7. 12131*: 3.0
8. 15544*: 3.1
9. 16627*: 3.1
10. 17131*: 3.1
11. 18555*: 3.1
12. 19187*: 3.0
13. 19651*: 3.0
14. 20276*: 3.0
15. 20568*: 3.0
16. 21224*: 3.0
17. 21668*: 3.1
18. 22395*: 3.1
19. 22443*: 3.1
20. 22711*: 3.0
21. 23428*: 3.0
22. 23683*: 3.1
23. 24340*: 3.0
24. 24884*: 3.1
25. 24904*: 3.0
26. 25447: 3.0
27. 26139: 3.0
28. 26760*: 3.0
29. 27156*: 5.0
30. 27355: 3.0
31. 27640*: 3.1
32. 27991: 3.0
33. 28031: 3.0
34. 28279: 3.1
35. 28759: 3.0
36. 31271: 3.1
37. 31639: 3.0
38. 31908*: 5.0
39. 31999: 3.0
40. 32968*: 3.0
41. 34027: 3.1
42. 34088*: 3.0
43. 34507: 3.0
44. 34867: 3.1
45. 35367: 3.0
46. 35539: 3.1
47. 36276*: 3.0
48. 36807: 3.0
49. 37219: 3.0
50. 37540*: 3.0
51. 37988*: 3.1
52. 39736*: 3.1
53. 39819: 3.0
54. 40299: 3.0
55. 40692*: 3.0
56. 41015: 3.0
57. 41063: 3.0
58. 41583: 3.0
59. 41671: 3.0
60. 42423: 3.0
61. 42619: 3.1
62. 42859: 3.1
63. 43192*: 3.0
64. 43307: 3.0
65. 43827: 3.0
66. 43847: 3.1
67. 44004*: 3.0
68. 45835: 3.0
69. 45887: 3.1
70. 46551: 3.0
71. 46587: 3.0
72. 48052*: 3.0
73. 48472*: 3.1
74. 48667: 3.1
75. 49128*: 3.0
76. 49812*: 3.0
77. 49924*: 3.0
78. 50739: 3.0
79. 50855: 3.0
80. 50983: 3.1
81. 51348*: 3.1
82. 51995: 3.0
83. 53839: 3.0
84. 53843: 3.1
85. 54071: 3.0
86. 54195: 3.0
87. 54251: 3.0
88. 54319: 3.1
89. 55247: 3.0
90. 55271: 3.0
91. 55623: 3.0
92. 57079: 3.0
93. 58920*: 3.1
94. 60099: 3.0
95. 60196*: 3.1
96. 60895: 3.1
97. 63079: 3.0
98. 63103: 3.1
99. 63303: 3.1
100. 64196*: 3.0
101. 64952*: 3.1
102. 65051: 3.1
103. 65203: 3.1
104. 65204*: 3.1
105. 65407: 3.1
106. 67480*: 5.0
107. 68584*: 3.1
108. 70244*: 3.0
109. 72435: 3.0
110. 72591: 3.1
111. 73007: 3.0
112. 73448*: 3.1
113. 73731: 3.0
114. 75847: 3.0
115. 77144*: 3.0
116. 78180*: 3.1
117. 78708*: 3.0
118. 79163: 3.1
119. 80516*: 3.1
120. 81867: 3.0
121. 83395: 3.0
122. 84072*: 3.0
123. 85199: 3.0
124. 85796*: 3.0
125. 86551: 3.0
126. 87503: 3.0
127. 87720*: 3.0
128. 87727: 3.0
129. 88447: 3.0
130. 89924*: 3.1
131. 90163: 3.1
132. 91471: 3.0
133. 91643: 3.0
134. 91860*: 3.0
135. 92660*: 3.1
136. 92712*: 3.0
137. 92827: 3.1
138. 93067: 3.0
139. 93207: 3.1
140. 93823: 3.0
141. 94420*: 3.0
142. 95155: 3.0
143. 95448*: 3.1
144. 95691: 3.0
145. 96436: 3.0
146. 96551: 3.0
147. 96827: 5.0
148. 97063: 3.0
149. 97555: 3.0
150. 97583: 3.0
151. 97687: 3.1
152. 97799: 3.0
153. 98347: 3.0
154. 98795: 3.0
155. 99707: 3.0
156. 99939: 3.0

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Statistics:

3.0: 101 times
3.1: 51 times
5.0: 4 times (27156,31908,67480,96827)

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 References: [1] Hideo Wada, A table of ideal class groups of imaginary quadratic fields, Proc. Japan Acad. 46 (1970), 401-403. [2] James R. Brink, The class field tower for imaginary quadratic number fields of type (3,3), Dissertation, Ohio State Univ., 1984. [3] G. W. Fung and H. C. Williams, On the computation of a table of complex cubic fields with discriminant D > -106, Math. Comp. 55 (1990), no. 191, 313-325. [4] H. C. Williams, Table errata, Math. Comp. 63 (1994), no. 207, 433. [5] Daniel C. Mayer, Two-Stage Towers of 3-Class Fields over Quadratic Fields, Univ. Graz, 2006. [6] Daniel C. Mayer, 3-Capitulation over Quadratic Fields with Discriminant |d| < 3*105 and 3-Class Group of Type (3,3), Univ. Graz, Computer Centre, 2006.