Memorial 2009



Dedicated to the Memory of

Alexander Aigner

in the Year 2009:

Quadruplets of Totally Real Cubic Number Fields

Section 9. All totally real cubic fields L with discriminant 800000 < d < 900000 and multiplicity m = 4

Breaking through beyond Ennola and Turunen's domain [1]

A single real quadratic field with four unramified cyclic cubic extensions
of principal factorization type Alpha 1 was found in this ninth range of length 100000.
We discovered and analyzed it on November 26, 2009, [2].

In this unexplored range, an eighth, ninth, and tenth unexpected and surprising result occurred (green color).
Three 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: (1,1,4,4), resp. D.5: (4,2,2,4),
up to now only known for complex quadratic base fields.
(Discovered and analyzed [2] on November 29, 2009, resp. December 04, 2009, resp. December 06, 2009.)

Continuation

Counter n Discriminant d Regulators R and class numbers h as pairs (R, h) Capitulation type
111 801368 (64.9, 9) (76.4, 6) (148.0, 3) (165.2, 3) a.1: (0,0,0,0)
112 804648 (218.6, 3) (219.9, 3) (229.4, 3) (250.8, 3) a.2: (0,0,0,4)
113 807937 (29.1, 3) (36.9, 3) (83.4, 3) (262.2, 3) a.3*: (0,0,1,0)
114 810661 (13.6, 15) (60.0, 3) (88.4, 3) (162.7, 3) D.10: (4,2,2,3)
115 814021 (52.7, 3) (67.7, 3) (107.1, 3) (194.5, 3) a.3: (0,0,4,0)
116 823512 (38.8, 15) (188.1, 3) (209.4, 3) (279.0, 3) a.3: (0,0,4,0)
117 829813 (44.3, 6) (58.0, 3) (68.8, 3) (212.1, 3) a.3: (2,0,0,0)
118 831484 (60.8, 3) (75.9, 3) (193.8, 3) (249.5, 3) a.2: (1,0,0,0)
119 835853 (56.6, 6) (100.6, 3) (105.9, 3) (148.6, 3) D.5: (1,1,4,4)
120 836493 (84.7, 6) (158.5, 3) (167.2, 3) (182.2, 3) a.2: (0,0,3,0)
121 859064 (141.2, 3) (141.9, 3) (149.3, 3) (243.4, 3) D.5: (4,2,2,4)
122 873969 (51.2, 3) (63.7, 3) (103.8, 3) (345.4, 3) a.3: (0,0,4,0)
123 874684 (99.1, 3) (99.6, 3) (178.2, 3) (285.4, 3) a.3: (2,0,0,0)
124 881689 (23.0, 3) (41.5, 3) (115.0, 3) (125.7, 3) a.3: (0,3,0,0)
125 893029 (27.3, 6) (61.1, 3) (137.6, 3) (157.3, 3) a.3*: (0,1,0,0)
126 893689 (76.9, 3) (98.3, 3) (124.8, 3) (468.7, 3) a.2: (1,0,0,0)


References:

[1] V. Ennola and R. Turunen,
On totally real cubic fields,
Math. Comp. 44 (1985), no. 170, 495-518.

[2] Daniel C. Mayer,
3-Capitulation over Quadratic Fields
with Discriminant |d| < 106 and 3-Class Group of Type (3,3)
,
(Latest Update)
Univ. Graz, Computer Centre, 2009.

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