Breaking through beyond Ennola and Turunen's domain [1]:
All totally real cubic fields L with discriminant 500000 < d < 600000 and multiplicity m = 4
Three real quadratic fields with four unramified cyclic cubic extensions
of principal factorization type Alpha 1 were found in this sixth range of length 100000.
(Discovered [2] on January 01, 2008, resp. October 02 and 05, 2009.)
In this unexplored range,
some totally unexpected and surprising results
occurred (green color):
d=534824 is the smallest discriminant where
type c.18 appears with non-terminal group G=Gal(K2|K) in CBF2a(5,6).
(Discovered [2] on August 20, 2009.)
d=540365 is the smallest discriminant where
type c.21 appears with non-terminal group G=Gal(K2|K) in CBF2a(5,6).
(Discovered [2] on January 01, 2008.)
Further, a fourth and fifth
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 type turned out to be D.10: (4,3,3,2), resp. D.10: (3,2,2,1),
up to now only known for complex quadratic base fields.
(Discovered and analyzed [2] on September 04, 2009, resp. September 30, 2009.)
Continuation
| Counter n | Discriminant d | Regulators R and class numbers h as pairs (R, h) | Capitulation type | |||
|---|---|---|---|---|---|---|
| 59 | 502796 | (113.5, 3) | (120.4, 3) | (129.6, 3) | (199.6, 3) | D.10: (4,3,3,2) |
| 60 | 510337 | (24.3, 3) | (24.7, 3) | (103.7, 3) | (188.4, 3) | a.3: (0,3,0,0) |
| 61 | 527068 | (87.6, 3) | (100.5, 3) | (102.0, 3) | (237.6, 3) | a.3*: (0,0,2,0) |
| 62 | 531437 | (27.7, 9) | (75.2, 3) | (103.3, 3) | (104.8, 3) | a.1: (0,0,0,0) |
| 63 | 531445 | (33.4, 6) | (34.6, 6) | (39.8, 6) | (193.4, 3) | a.3: (0,0,0,3) |
| 64 | 534824 | (34.3, 9) | (89.2, 3) | (123.3, 3) | (196.6, 3) | c.18: (0,3,1,3) |
| 65 | 535441 | (16.6, 3) | (21.5, 3) | (39.4, 3) | (254.6, 3) | a.3*: (2,0,0,0) |
| 66 | 540365 | (39.1, 9) | (116.3, 3) | (118.9, 3) | (131.6, 3) | c.21: (0,2,3,1) |
| 67 | 548549 | (92.0, 3) | (105.1, 3) | (107.3, 3) | (150.2, 3) | a.2: (0,0,0,4) |
| 68 | 549133 | (50.8, 3) | (66.4, 3) | (77.1, 3) | (161.7, 3) | a.3: (0,0,0,2) |
| 69 | 551384 | (90.8, 3) | (118.9, 3) | (129.2, 3) | (163.4, 3) | a.3*: (4,0,0,0) |
| 70 | 551692 | (50.0, 6) | (65.7, 3) | (98.4, 3) | (109.2, 6) | a.2: (0,0,0,4) |
| 71 | 552392 | (103.4, 3) | (117.3, 3) | (150.2, 3) | (164.3, 3) | a.2: (0,2,0,0) |
| 72 | 557657 | (35.6, 3) | (44.0, 3) | (66.7, 3) | (130.3, 6) | a.3: (0,0,2,0) |
| 73 | 567473 | (14.3, 6) | (41.6, 3) | (51.3, 3) | (256.7, 3) | a.3*: (3,0,0,0) |
| 74 | 575729 | (42.9, 3) | (44.9, 3) | (72.4, 3) | (256.7, 3) | D.10: (3,2,2,1) |
| 75 | 578581 | (84.2, 3) | (96.1, 3) | (96.2, 3) | (218.9, 3) | a.3: (0,4,0,0) |
| 76 | 586760 | (53.5, 9) | (94.9, 6) | (161.1, 3) | (166.1, 3) | a.1: (0,0,0,0) |
| 77 | 593941 | (40.7, 3) | (49.1, 3) | (97.4, 3) | (148.2, 3) | a.3: (0,0,4,0) |
| 78 | 595009 | (25.4, 3) | (50.2, 3) | (66.0, 9) | (78.7, 3) | a.1: (0,0,0,0) |
| 79 | 597068 | (137.4, 3) | (141.7, 3) | (144.2, 3) | (157.6, 3) | a.3: (3,0,0,0) |