Memorial 2009



Dedicated to the Memory of

Alexander Aigner

in the Year 2009:

Quadruplets of Totally Real Cubic Number Fields

Section 6. All totally real cubic fields L with discriminant 500000 < d < 600000 and multiplicity m = 4

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

Three real quadratic fields with four unramified cyclic cubic extensions
of principal factorization type Alpha 1 were found in this sixth range of length 100000.
(Discovered [2] on January 01, 2008, resp. October 02 and 05, 2009.)

In this unexplored range, some totally unexpected and surprising results occurred (green color):
d=534824 is the smallest discriminant where
type c.18 appears with non-terminal group G=Gal(K2|K) in CBF2a(5,6).
(Discovered [2] on August 20, 2009.)
d=540365 is the smallest discriminant where
type c.21 appears with non-terminal group G=Gal(K2|K) in CBF2a(5,6).
(Discovered [2] on January 01, 2008.)

Further, a fourth and fifth 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 type turned out to be D.10: (4,3,3,2), resp. D.10: (3,2,2,1),
up to now only known for complex quadratic base fields.
(Discovered and analyzed [2] on September 04, 2009, resp. September 30, 2009.)

Continuation

Counter n Discriminant d Regulators R and class numbers h as pairs (R, h) Capitulation type
59 502796 (113.5, 3) (120.4, 3) (129.6, 3) (199.6, 3) D.10: (4,3,3,2)
60 510337 (24.3, 3) (24.7, 3) (103.7, 3) (188.4, 3) a.3: (0,3,0,0)
61 527068 (87.6, 3) (100.5, 3) (102.0, 3) (237.6, 3) a.3*: (0,0,2,0)
62 531437 (27.7, 9) (75.2, 3) (103.3, 3) (104.8, 3) a.1: (0,0,0,0)
63 531445 (33.4, 6) (34.6, 6) (39.8, 6) (193.4, 3) a.3: (0,0,0,3)
64 534824 (34.3, 9) (89.2, 3) (123.3, 3) (196.6, 3) c.18: (0,3,1,3)
65 535441 (16.6, 3) (21.5, 3) (39.4, 3) (254.6, 3) a.3*: (2,0,0,0)
66 540365 (39.1, 9) (116.3, 3) (118.9, 3) (131.6, 3) c.21: (0,2,3,1)
67 548549 (92.0, 3) (105.1, 3) (107.3, 3) (150.2, 3) a.2: (0,0,0,4)
68 549133 (50.8, 3) (66.4, 3) (77.1, 3) (161.7, 3) a.3: (0,0,0,2)
69 551384 (90.8, 3) (118.9, 3) (129.2, 3) (163.4, 3) a.3*: (4,0,0,0)
70 551692 (50.0, 6) (65.7, 3) (98.4, 3) (109.2, 6) a.2: (0,0,0,4)
71 552392 (103.4, 3) (117.3, 3) (150.2, 3) (164.3, 3) a.2: (0,2,0,0)
72 557657 (35.6, 3) (44.0, 3) (66.7, 3) (130.3, 6) a.3: (0,0,2,0)
73 567473 (14.3, 6) (41.6, 3) (51.3, 3) (256.7, 3) a.3*: (3,0,0,0)
74 575729 (42.9, 3) (44.9, 3) (72.4, 3) (256.7, 3) D.10: (3,2,2,1)
75 578581 (84.2, 3) (96.1, 3) (96.2, 3) (218.9, 3) a.3: (0,4,0,0)
76 586760 (53.5, 9) (94.9, 6) (161.1, 3) (166.1, 3) a.1: (0,0,0,0)
77 593941 (40.7, 3) (49.1, 3) (97.4, 3) (148.2, 3) a.3: (0,0,4,0)
78 595009 (25.4, 3) (50.2, 3) (66.0, 9) (78.7, 3) a.1: (0,0,0,0)
79 597068 (137.4, 3) (141.7, 3) (144.2, 3) (157.6, 3) a.3: (3,0,0,0)


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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