Breaking through beyond Ennola and Turunen's domain [1] and reaching the target:
All totally real cubic fields L with discriminant 900000 < d < 106 and multiplicity m = 4
Two real quadratic fields with four unramified cyclic cubic extensions
of principal factorization type Alpha 1 were found in this tenth range of length 100000.
We discovered and analyzed them on December 17, 2009, [2].
Further, an eleventh and twelfth
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
discovered and analyzed on December 17, 2009, [2].
The capitulation type turned out to be G.19: (4,3,2,1), resp. H.4: (4,4,4,1),
up to now only known for complex quadratic base fields.
d=945813 is a second discriminant where
type c.21 appears with non-terminal group G=Gal(K2|K) in CBF2a(5,6).
(Discovered and analyzed [2] on December 17, 2009.)
Navigation
| Counter n | Discriminant d | Regulators R and class numbers h as pairs (R, h) | Capitulation type | |||
|---|---|---|---|---|---|---|
| 127 | 902333 | (101.5, 3) | (108.8, 3) | (118.9, 3) | (140.2, 3) | a.3: (0,0,0,1) |
| 128 | 907629 | (108.4, 3) | (120.6, 3) | (154.7, 3) | (198.8, 3) | a.2: (0,0,3,0) |
| 129 | 907709 | (49.3, 6) | (87.9, 6) | (98.7, 3) | (120.6, 3) | a.2: (0,0,3,0) |
| 130 | 908241 | (89.1, 3) | (109.5, 3) | (120.3, 3) | (534.0, 3) | a.3: (0,0,0,3) |
| 131 | 916181 | (116.9, 3) | (119.2, 3) | (120.3, 3) | (189.9, 3) | a.3*: (0,4,0,0) |
| 132 | 935665 | (23.4, 6) | (36.7, 3) | (38.8, 3) | (633.1, 3) | a.3: (0,0,1,0) |
| 133 | 939569 | (35.7, 3) | (53.1, 3) | (73.9, 3) | (289.7, 3) | a.3: (0,4,0,0) |
| 134 | 940593 | (32.3, 9) | (61.2, 3) | (74.8, 3) | (406.5, 3) | a.1: (0,0,0,0) |
| 135 | 942961 | (30.3, 3) | (45.0, 12) | (51.9, 3) | (85.4, 3) | a.3*: (3,0,0,0) |
| 136 | 943077 | (22.9, 21) | (125.8, 3) | (146.0, 3) | (194.9, 3) | G.19: (4,3,2,1) |
| 137 | 944760 | (249.2, 3) | (280.3, 3) | (290.7, 3) | (298.7, 3) | a.3: (0,0,4,0) |
| 138 | 945813 | (55.2, 9) | (145.0, 3) | (156.7, 3) | (212.0, 3) | c.21: (0,2,3,1) |
| 139 | 957013 | (41.5, 6) | (68.1, 3) | (108.7, 3) | (209.8, 3) | H.4: (2,1,2,2) |
| 140 | 957484 | (101.2, 3) | (106.1, 3) | (180.6, 3) | (306.6, 3) | a.2: (0,2,0,0) |
| 141 | 959629 | (69.9, 3) | (76.2, 3) | (125.1, 3) | (203.7, 3) | a.2: (0,0,0,4) |
| 142 | 966053 | (113.6, 3) | (128.2, 3) | (131.2, 3) | (159.7, 3) | a.3*: (4,0,0,0) |
| 143 | 966489 | (19.5, 9) | (60.6, 3) | (68.0, 3) | (566.8, 3) | a.1: (0,0,0,0) |
| 144 | 967928 | (180.5, 3) | (198.1, 3) | (198.5, 3) | (211.1, 3) | a.3*: (0,0,2,0) |
| 145 | 974157 | (143.5, 3) | (143.8, 3) | (157.2, 3) | (208.6, 3) | a.3*: (0,0,0,2) |
| 146 | 980108 | (96.4, 6) | (170.1, 3) | (175.0, 3) | (213.2, 3) | a.2: (0,0,0,4) |
| 147 | 982049 | (24.9, 6) | (45.3, 3) | (46.1, 6) | (274.3, 3) | a.3*: (0,0,2,0) |
| 148 | 993349 | (70.1, 3) | (86.8, 3) | (129.8, 3) | (203.4, 3) | a.2: (0,0,0,4) |
| 149 | 994008 | (103.8, 6) | (185.0, 3) | (252.9, 3) | (298.0, 3) | a.2: (1,0,0,0) |