Super Computing Centre 2006



Discriminants < 106 of Real Quadratic Fields

with 3-Class Rank 2





Quadratic 3-Class Groups via Cubic Fields.

In April 2006 we computed all totally real cubic fields with fundamental discriminants 0 < d < 106,
i. e., whose normal fields are unramified over their quadratic subfields.

The quadruplets of non-isomorphic fields sharing the same discriminant among these totally real cubic fields
are in one to one correspondence with the 161 real quadratic fields of 3-class rank 2.
Among the latter we have 149 with 3-class group of type (3,3) and 12 with 3-class group of type (9,3).

Similarly as we did on page 77 in our 1991 paper [5] for complex quadratic base fields
we suggest now for real quadratic base fields:
"It would be interesting to investigate the principalization
in the unramified cyclic cubic extensions of all these quadratic fields.
This would show more reliable tendencies in the population of the various principalization types."

We have started this extensive Real Capitulation Project on January 16, 2006 (after preparations since January 04, 2006),
and completed it on December 17, 2009, with extensive computational results [8] and deep theoretical insight [7].

1. The 149 Discriminants with 3-Class Group of Type (3,3)

1. 32009 2. 42817 3. 62501 4. 72329 5. 94636 6. 103809 7. 114889 8. 130397 9. 142097 10. 151141

11. 152949 12. 153949 13. 172252 14. 173944 15. 184137 16. 189237 17. 206776 18. 209765 19. 213913 20. 214028
21. 214712 22. 219461 23. 220217 24. 250748 25. 252977 26. 259653 27. 265245 28. 275881 29. 283673 30. 298849
31. 320785 32. 321053 33. 326945 34. 333656 35. 335229 36. 341724 37. 342664 38. 358285 39. 363397 40. 371965
41. 390876 42. 400369 43. 412277 44. 415432 45. 422573 46. 424236 47. 431761 48. 449797 49. 459964 50. 460817
51. 468472 52. 471057 53. 471713 54. 476124 55. 476152 56. 486221 57. 486581 58. 494236 59. 502796 60. 510337
61. 527068 62. 531437 63. 531445 64. 534824 65. 535441 66. 540365 67. 548549 68. 549133 69. 551384 70. 551692
71. 552392 72. 557657 73. 567473 74. 575729 75. 578581 76. 586760 77. 593941 78. 595009 79. 597068 80. 600085
81. 602521 82. 621429 83. 621749 84. 626441 85. 631769 86. 636632 87. 637820 88. 654796 89. 665832 90. 681276
91. 686977 92. 689896 93. 698556 94. 710652 95. 718705 96. 719105 97. 722893 98. 726933 99. 729293 100. 747496

101. 750376 102. 751657 103. 775480 104. 775661 105. 781177 106. 782737 107. 782876 108. 784997 109. 785269 110. 790085
111. 801368 112. 804648 113. 807937 114. 810661 115. 814021 116. 823512 117. 829813 118. 831484 119. 835853 120. 836493
121. 859064 122. 873969 123. 874684 124. 881689 125. 893029 126. 893689 127. 902333 128. 907629 129. 907709 130. 908241
131. 916181 132. 935665 133. 939569 134. 940593 135. 942961 136. 943077 137. 944760 138. 945813 139. 957013 140. 957484
141. 959629 142. 966053 143. 966489 144. 967928 145. 974157 146. 980108 147. 982049 148. 993349 149. 994008

2. The 12 Discriminants with 3-Class Group of Type (9,3)

1. 255973 2. 282461 3. 384369 4. 529393 5. 540213 6. 626264 7. 635909 8. 700313 9. 783689 10. 895449
11. 946733 12. 948777



References:

[1] Arnold Scholz und Olga Taussky,
Die Hauptideale der kubischen Klassenkörper
imaginär quadratischer Zahlkörper,
J. reine angew. Math.171 (1934), 19 - 41.

[2] Franz-Peter Heider und Bodo Schmithals,
Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen,
J. reine angew. Math. 336 (1982), 1 - 25.

[3] James R. Brink,
The class field tower for imaginary quadratic number fields of type (3,3),
Dissertation, Ohio State Univ., 1984.

[4] Brigitte Nebelung,
Klassifikation metabelscher 3-Gruppen
mit Faktorkommutatorgruppe vom Typ (3,3)
und Anwendung auf das Kapitulationsproblem,
Inauguraldissertation, Köln, 1989.

[5] Daniel C. Mayer,
Principalization in complex S3-fields,
Congressus Numerantium 80 (1991), 73 - 87.

[6] Daniel C. Mayer,
List of discriminants dL<200000 of totally real cubic fields L,
arranged according to their multiplicities m and conductors f,
Dept. of Comp. Sci., Univ. of Manitoba, 1991.

[7] Daniel C. Mayer,
Two-Stage Towers of 3-Class Fields over Quadratic Fields,
Univ. Graz, 2006.

[8] 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.

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