# Quadruplets of Totally Real Cubic Number Fields

### Reaching the border of Ennola and Turunen's domain [1]

In this range, d=494236 is the smallest discriminant where
type a.3 appears in its first excited state with group G=Gal(K2|K) in CF2a(6).
(Discovered [2] on August 16, 2007, analyzed on February 04, 2008, independently from [1].)

Further, a third unexpected and surprising result occurred (green color).
A single real quadratic field with four unramified cyclic cubic extensions
of principal factorization type Delta 1 has been found.
The capitulation type turned out to be D.10: (1,3,4,1),
up to now only known for complex quadratic base fields.
(Discovered [2] on April 28, 2006, analyzed on June 13, 2006, independently from [1].)

Continuation

Counter n Discriminant d Regulators R and class numbers h as pairs (R, h) Capitulation type
42 400369 (13.0, 3) (13.6, 3) (57.9, 3) (229.2, 3) a.2: (0,0,0,4)
43 412277 (21.1, 12) (77.3, 3) (77.8, 3) (85.3, 3) a.3*: (0,0,0,3)
44 415432 (65.3, 3) (67.3, 3) (93.5, 3) (215.8, 3) a.3: (0,0,2,0)
45 422573 (69.9, 3) (72.4, 3) (78.4, 3) (101.1, 3) D.10: (1,3,4,1)
46 424236 (126.0, 3) (127.0, 3) (165.6, 3) (199.8, 3) a.3*: (0,0,1,0)
47 431761 (18.1, 3) (30.1, 3) (52.4, 3) (162.3, 3) a.2: (0,0,3,0)
48 449797 (13.9, 12) (55.7, 3) (62.8, 3) (152.8, 3) a.3: (0,0,1,0)
49 459964 (52.0, 3) (62.1, 3) (146.0, 3) (174.5, 3) a.3*: (0,0,1,0)
50 460817 (37.0, 3) (41.6, 3) (52.3, 3) (246.3, 3) a.2: (1,0,0,0)
51 468472 (72.7, 3) (74.1, 3) (95.9, 6) (114.9, 3) a.3: (3,0,0,0)
52 471057 (46.0, 3) (54.90, 3) (54.95, 3) (301.6, 3) a.3: (3,0,0,0)
53 471713 (38.8, 3) (39.8, 3) (57.0, 3) (232.4, 3) a.3*: (0,0,2,0)
54 476124 (140.6, 3) (153.1, 3) (161.4, 3) (230.6, 3) a.3: (0,4,0,0)
55 476152 (94.2, 3) (95.4, 3) (102.8, 3) (243.9, 3) a.3*: (0,0,4,0)
56 486221 (59.5, 3) (74.1, 3) (87.1, 3) (117.5, 3) a.3: (0,0,0,3)
57 486581 (29.6, 6) (73.2, 3) (86.2, 3) (104.4, 3) a.2: (1,0,0,0)
58 494236 (36.4, 9) (60.1, 3) (84.8, 3) (178.4, 3) a.3/V.1: (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.