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)
|