Breaking through beyond Ennola and Turunen's domain [1]:
All totally real cubic fields L with discriminant 700000 < d < 800000 and multiplicity m = 4
In this unexplored range, a
totally unexpected and surprising result
occurred (green color).
A real quadratic field K with
capitulation type b.10: (0,0,4,3) has been found.
710652 is the smallest discriminant, where
the group G=Gal(K2|K) is of lower than second maximal class.
(Discovered and analyzed [2] on November 09, 2009, independently from [1].)
Further, in this range, d=790085 is the smallest discriminant where
type a.2 appears in its first excited state with group G=Gal(K2|K) in CF2a(6).
(Discovered and analyzed [2] on November 25, 2009, independently from [1].)
Continuation
| Counter n | Discriminant d | Regulators R and class numbers h as pairs (R, h) | Capitulation type | |||
|---|---|---|---|---|---|---|
| 94 | 710652 | (35.2, 18) | (68.0, 9) | (198.7, 3) | (222.0, 3) | b.10: (0,0,4,3) |
| 95 | 718705 | (27.9, 6) | (30.9, 3) | (36.3, 3) | (526.1, 3) | a.3: (0,0,0,3) |
| 96 | 719105 | (47.3, 3) | (56.1, 3) | (74.7, 3) | (303.3, 3) | a.3: (0,1,0,0) |
| 97 | 722893 | (33.0, 6) | (73.0, 3) | (83.3, 3) | (198.9, 3) | a.3: (0,0,0,2) |
| 98 | 726933 | (45.9, 9) | (135.6, 3) | (156.0, 3) | (176.2, 3) | a.1: (0,0,0,0) |
| 99 | 729293 | (100.5, 3) | (110.4, 3) | (111.7, 3) | (127.7, 3) | a.3*: (0,0,2,0) |
| 100 | 747496 | (78.0, 3) | (83.7, 3) | (100.9, 3) | (354.0, 3) | a.3*: (0,0,2,0) |
| 101 | 750376 | (105.2, 3) | (132.0, 3) | (153.8, 3) | (255.8, 3) | a.3*: (2,0,0,0) |
| 102 | 751657 | (11.4, 6) | (22.5, 3) | (61.0, 6) | (197.9, 3) | a.3: (0,0,0,3) |
| 103 | 775480 | (109.9, 3) | (123.9, 3) | (150.7, 3) | (336.6, 3) | a.2: (0,0,0,4) |
| 104 | 775661 | (52.0, 6) | (88.0, 3) | (91.0, 3) | (154.0, 3) | a.2: (0,0,3,0) |
| 105 | 781177 | (15.0, 6) | (26.6, 3) | (38.2, 3) | (262.5, 6) | a.2: (0,2,0,0) |
| 106 | 782737 | (23.4, 3) | (41.4, 3) | (91.0, 3) | (177.5, 3) | a.3*: (0,1,0,0) |
| 107 | 782876 | (133.5, 3) | (152.9, 3) | (155.4, 3) | (228.1, 3) | a.2: (1,0,0,0) |
| 108 | 784997 | (106.0, 3) | (109.2, 3) | (113.4, 3) | (131.1, 3) | a.3*: (0,0,0,1) |
| 109 | 785269 | (125.9, 3) | (180.4, 3) | (218.1, 3) | (240.4, 3) | a.3: (3,0,0,0) |
| 110 | 790085 | (52.4, 9) | (104.5, 3) | (123.6, 3) | (133.9, 3) | a.2/V.1: (1,0,0,0) |