Breaking through beyond Ennola and Turunen's domain [1]:
All totally real cubic fields L with discriminant 600000 < d < 700000 and multiplicity m = 4
This is the first range, where none of the 3-class numbers of the
absolute non-galois cubic subfields of the
unramified cyclic cubic extension fields is divisible by 9,
and all 3-class field towers terminate with the second stage.
In this unexplored range, a sixth and seventh
unexpected and surprising result
occurred (green color).
Two 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: (3,4,3,4),
up to now only known for complex quadratic base fields.
(Discovered and analyzed [2] on October 29, 2009, resp. October 30, 2009.)
Continuation
| Counter n | Discriminant d | Regulators R and class numbers h as pairs (R, h) | Capitulation type | |||
|---|---|---|---|---|---|---|
| 80 | 600085 | (68.4, 3) | (75.1, 3) | (84.8, 3) | (206.4, 3) | a.3: (3,0,0,0) |
| 81 | 602521 | (38.1, 3) | (47.5, 15) | (47.6, 3) | (117.6, 3) | a.2: (0,0,0,4) |
| 82 | 621429 | (22.7, 12) | (92.2, 3) | (150.1, 3) | (159.3, 3) | a.3: (4,0,0,0) |
| 83 | 621749 | (68.2, 3) | (85.3, 3) | (97.0, 3) | (120.1, 3) | a.3*: (0,0,2,0) |
| 84 | 626411 | (43.5, 3) | (58.3, 3) | (67.0, 3) | (115.2, 6) | D.10: (4,2,2,3) |
| 85 | 631769 | (18.8, 12) | (41.9, 3) | (43.9, 3) | (50.0, 15) | D.5: (3,4,3,4) |
| 86 | 636632 | (137.2, 3) | (148.3, 3) | (150.5, 3) | (165.1, 3) | a.3: (0,0,0,1) |
| 87 | 637820 | (150.0, 3) | (161.7, 3) | (169.2, 3) | (204.1, 3) | a.3*: (0,4,0,0) |
| 88 | 654796 | (89.1, 3) | (99.9, 3) | (126.8, 3) | (201.2, 3) | a.3: (0,4,0,0) |
| 89 | 665832 | (200.4, 3) | (202.9, 3) | (206.9, 3) | (211.0, 3) | a.3: (0,0,4,0) |
| 90 | 681276 | (131.9, 3) | (158.8, 3) | (206.0, 3) | (254.7, 3) | a.3*: (0,0,2,0) |
| 91 | 686977 | (19.1, 3) | (30.0, 3) | (74.1, 3) | (225.9, 3) | a.3*: (0,0,1,0) |
| 92 | 689896 | (199.3, 3) | (233.2, 3) | (303.5, 3) | (314.1, 3) | a.3: (0,3,0,0) |
| 93 | 698556 | (68.6, 12) | (147.8, 3) | (177.0, 3) | (191.7, 3) | a.2: (1,0,0,0) |