Reaching the border of Ennola and Turunen's domain [1]:
All totally real cubic fields L with discriminant 400000 < d < 500000 and multiplicity m = 4
In this range, d=494236 is the smallest discriminant where
type a.3 appears in its first excited state with group G=Gal(K2|K) in CF2a(6).
(Discovered [2] on August 16, 2007, analyzed on February 04, 2008, independently from [1].)
Further, a third
unexpected and surprising result
occurred (green color).
A single real quadratic field with four unramified cyclic cubic extensions
of principal factorization type Delta 1 has been found.
The capitulation type turned out to be D.10: (1,3,4,1),
up to now only known for complex quadratic base fields.
(Discovered [2] on April 28, 2006, analyzed on June 13, 2006, independently from [1].)
Continuation
| Counter n | Discriminant d | Regulators R and class numbers h as pairs (R, h) | Capitulation type | |||
|---|---|---|---|---|---|---|
| 42 | 400369 | (13.0, 3) | (13.6, 3) | (57.9, 3) | (229.2, 3) | a.2: (0,0,0,4) |
| 43 | 412277 | (21.1, 12) | (77.3, 3) | (77.8, 3) | (85.3, 3) | a.3*: (0,0,0,3) |
| 44 | 415432 | (65.3, 3) | (67.3, 3) | (93.5, 3) | (215.8, 3) | a.3: (0,0,2,0) |
| 45 | 422573 | (69.9, 3) | (72.4, 3) | (78.4, 3) | (101.1, 3) | D.10: (1,3,4,1) |
| 46 | 424236 | (126.0, 3) | (127.0, 3) | (165.6, 3) | (199.8, 3) | a.3*: (0,0,1,0) |
| 47 | 431761 | (18.1, 3) | (30.1, 3) | (52.4, 3) | (162.3, 3) | a.2: (0,0,3,0) |
| 48 | 449797 | (13.9, 12) | (55.7, 3) | (62.8, 3) | (152.8, 3) | a.3: (0,0,1,0) |
| 49 | 459964 | (52.0, 3) | (62.1, 3) | (146.0, 3) | (174.5, 3) | a.3*: (0,0,1,0) |
| 50 | 460817 | (37.0, 3) | (41.6, 3) | (52.3, 3) | (246.3, 3) | a.2: (1,0,0,0) |
| 51 | 468472 | (72.7, 3) | (74.1, 3) | (95.9, 6) | (114.9, 3) | a.3: (3,0,0,0) |
| 52 | 471057 | (46.0, 3) | (54.90, 3) | (54.95, 3) | (301.6, 3) | a.3: (3,0,0,0) |
| 53 | 471713 | (38.8, 3) | (39.8, 3) | (57.0, 3) | (232.4, 3) | a.3*: (0,0,2,0) |
| 54 | 476124 | (140.6, 3) | (153.1, 3) | (161.4, 3) | (230.6, 3) | a.3: (0,4,0,0) |
| 55 | 476152 | (94.2, 3) | (95.4, 3) | (102.8, 3) | (243.9, 3) | a.3*: (0,0,4,0) |
| 56 | 486221 | (59.5, 3) | (74.1, 3) | (87.1, 3) | (117.5, 3) | a.3: (0,0,0,3) |
| 57 | 486581 | (29.6, 6) | (73.2, 3) | (86.2, 3) | (104.4, 3) | a.2: (1,0,0,0) |
| 58 | 494236 | (36.4, 9) | (60.1, 3) | (84.8, 3) | (178.4, 3) | a.3/V.1: (3,0,0,0) |