# 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)

 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.