Breaking through beyond Ennola and Turunen's domain [1]:
All totally real cubic fields L with discriminant 600000 < d < 700000 and multiplicity m = 4
This is the first range, where none of the 3-class numbers of the
absolute non-galois cubic subfields of the
unramified cyclic cubic extension fields is divisible by 9,
and all 3-class field towers terminate with the second stage.
In this unexplored range, a sixth and seventh
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 types turned out to be D.10: (4,2,2,3), resp. D.5: (3,4,3,4),
up to now only known for complex quadratic base fields.
(Discovered and analyzed [2] on October 29, 2009, resp. October 30, 2009.)
Continuation
Counter n
|
Discriminant d
|
Regulators R and class numbers h as pairs (R, h)
|
Capitulation type
|
80
|
600085
|
(68.4, 3)
|
(75.1, 3)
|
(84.8, 3)
|
(206.4, 3)
|
a.3: (3,0,0,0)
|
81
|
602521
|
(38.1, 3)
|
(47.5, 15)
|
(47.6, 3)
|
(117.6, 3)
|
a.2: (0,0,0,4)
|
82
|
621429
|
(22.7, 12)
|
(92.2, 3)
|
(150.1, 3)
|
(159.3, 3)
|
a.3: (4,0,0,0)
|
83
|
621749
|
(68.2, 3)
|
(85.3, 3)
|
(97.0, 3)
|
(120.1, 3)
|
a.3*: (0,0,2,0)
|
84
|
626411
|
(43.5, 3)
|
(58.3, 3)
|
(67.0, 3)
|
(115.2, 6)
|
D.10: (4,2,2,3)
|
85
|
631769
|
(18.8, 12)
|
(41.9, 3)
|
(43.9, 3)
|
(50.0, 15)
|
D.5: (3,4,3,4)
|
86
|
636632
|
(137.2, 3)
|
(148.3, 3)
|
(150.5, 3)
|
(165.1, 3)
|
a.3: (0,0,0,1)
|
87
|
637820
|
(150.0, 3)
|
(161.7, 3)
|
(169.2, 3)
|
(204.1, 3)
|
a.3*: (0,4,0,0)
|
88
|
654796
|
(89.1, 3)
|
(99.9, 3)
|
(126.8, 3)
|
(201.2, 3)
|
a.3: (0,4,0,0)
|
89
|
665832
|
(200.4, 3)
|
(202.9, 3)
|
(206.9, 3)
|
(211.0, 3)
|
a.3: (0,0,4,0)
|
90
|
681276
|
(131.9, 3)
|
(158.8, 3)
|
(206.0, 3)
|
(254.7, 3)
|
a.3*: (0,0,2,0)
|
91
|
686977
|
(19.1, 3)
|
(30.0, 3)
|
(74.1, 3)
|
(225.9, 3)
|
a.3*: (0,0,1,0)
|
92
|
689896
|
(199.3, 3)
|
(233.2, 3)
|
(303.5, 3)
|
(314.1, 3)
|
a.3: (0,3,0,0)
|
93
|
698556
|
(68.6, 12)
|
(147.8, 3)
|
(177.0, 3)
|
(191.7, 3)
|
a.2: (1,0,0,0)
|