# Quadruplets of Totally Real Cubic Number Fields

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

In this unexplored range, a totally unexpected and surprising result occurred (green color).
A real quadratic field K with capitulation type b.10: (0,0,4,3) has been found.
710652 is the smallest discriminant, where
the group G=Gal(K2|K) is of lower than second maximal class.
(Discovered and analyzed [2] on November 09, 2009, independently from [1].)

Further, in this range, d=790085 is the smallest discriminant where
type a.2 appears in its first excited state with group G=Gal(K2|K) in CF2a(6).
(Discovered and analyzed [2] on November 25, 2009, independently from [1].)

Continuation

Counter n Discriminant d Regulators R and class numbers h as pairs (R, h) Capitulation type
94 710652 (35.2, 18) (68.0, 9) (198.7, 3) (222.0, 3) b.10: (0,0,4,3)
95 718705 (27.9, 6) (30.9, 3) (36.3, 3) (526.1, 3) a.3: (0,0,0,3)
96 719105 (47.3, 3) (56.1, 3) (74.7, 3) (303.3, 3) a.3: (0,1,0,0)
97 722893 (33.0, 6) (73.0, 3) (83.3, 3) (198.9, 3) a.3: (0,0,0,2)
98 726933 (45.9, 9) (135.6, 3) (156.0, 3) (176.2, 3) a.1: (0,0,0,0)
99 729293 (100.5, 3) (110.4, 3) (111.7, 3) (127.7, 3) a.3*: (0,0,2,0)
100 747496 (78.0, 3) (83.7, 3) (100.9, 3) (354.0, 3) a.3*: (0,0,2,0)
101 750376 (105.2, 3) (132.0, 3) (153.8, 3) (255.8, 3) a.3*: (2,0,0,0)
102 751657 (11.4, 6) (22.5, 3) (61.0, 6) (197.9, 3) a.3: (0,0,0,3)
103 775480 (109.9, 3) (123.9, 3) (150.7, 3) (336.6, 3) a.2: (0,0,0,4)
104 775661 (52.0, 6) (88.0, 3) (91.0, 3) (154.0, 3) a.2: (0,0,3,0)
105 781177 (15.0, 6) (26.6, 3) (38.2, 3) (262.5, 6) a.2: (0,2,0,0)
106 782737 (23.4, 3) (41.4, 3) (91.0, 3) (177.5, 3) a.3*: (0,1,0,0)
107 782876 (133.5, 3) (152.9, 3) (155.4, 3) (228.1, 3) a.2: (1,0,0,0)
108 784997 (106.0, 3) (109.2, 3) (113.4, 3) (131.1, 3) a.3*: (0,0,0,1)
109 785269 (125.9, 3) (180.4, 3) (218.1, 3) (240.4, 3) a.3: (3,0,0,0)
110 790085 (52.4, 9) (104.5, 3) (123.6, 3) (133.9, 3) a.2/V.1: (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.