Breaking through beyond Ennola and Turunen's domain [1]:
All totally real cubic fields L with discriminant 800000 < d < 900000 and multiplicity m = 4
A single real quadratic field with four unramified cyclic cubic extensions
of principal factorization type Alpha 1 was found in this ninth range of length 100000.
We discovered and analyzed it on November 26, 2009, [2].
In this unexplored range, an eighth, ninth, and tenth
unexpected and surprising result
occurred (green color).
Three real quadratic fields with four unramified cyclic cubic extensions
of principal factorization type Delta 1 have been found.
The capitulation types turned out to be D.10: (4,2,2,3), resp. D.5: (1,1,4,4), resp. D.5: (4,2,2,4),
up to now only known for complex quadratic base fields.
(Discovered and analyzed [2] on November 29, 2009, resp. December 04, 2009, resp. December 06, 2009.)
Continuation
| Counter n | Discriminant d | Regulators R and class numbers h as pairs (R, h) | Capitulation type | |||
|---|---|---|---|---|---|---|
| 111 | 801368 | (64.9, 9) | (76.4, 6) | (148.0, 3) | (165.2, 3) | a.1: (0,0,0,0) |
| 112 | 804648 | (218.6, 3) | (219.9, 3) | (229.4, 3) | (250.8, 3) | a.2: (0,0,0,4) |
| 113 | 807937 | (29.1, 3) | (36.9, 3) | (83.4, 3) | (262.2, 3) | a.3*: (0,0,1,0) |
| 114 | 810661 | (13.6, 15) | (60.0, 3) | (88.4, 3) | (162.7, 3) | D.10: (4,2,2,3) |
| 115 | 814021 | (52.7, 3) | (67.7, 3) | (107.1, 3) | (194.5, 3) | a.3: (0,0,4,0) |
| 116 | 823512 | (38.8, 15) | (188.1, 3) | (209.4, 3) | (279.0, 3) | a.3: (0,0,4,0) |
| 117 | 829813 | (44.3, 6) | (58.0, 3) | (68.8, 3) | (212.1, 3) | a.3: (2,0,0,0) |
| 118 | 831484 | (60.8, 3) | (75.9, 3) | (193.8, 3) | (249.5, 3) | a.2: (1,0,0,0) |
| 119 | 835853 | (56.6, 6) | (100.6, 3) | (105.9, 3) | (148.6, 3) | D.5: (1,1,4,4) |
| 120 | 836493 | (84.7, 6) | (158.5, 3) | (167.2, 3) | (182.2, 3) | a.2: (0,0,3,0) |
| 121 | 859064 | (141.2, 3) | (141.9, 3) | (149.3, 3) | (243.4, 3) | D.5: (4,2,2,4) |
| 122 | 873969 | (51.2, 3) | (63.7, 3) | (103.8, 3) | (345.4, 3) | a.3: (0,0,4,0) |
| 123 | 874684 | (99.1, 3) | (99.6, 3) | (178.2, 3) | (285.4, 3) | a.3: (2,0,0,0) |
| 124 | 881689 | (23.0, 3) | (41.5, 3) | (115.0, 3) | (125.7, 3) | a.3: (0,3,0,0) |
| 125 | 893029 | (27.3, 6) | (61.1, 3) | (137.6, 3) | (157.3, 3) | a.3*: (0,1,0,0) |
| 126 | 893689 | (76.9, 3) | (98.3, 3) | (124.8, 3) | (468.7, 3) | a.2: (1,0,0,0) |