# Quadruplets of Totally Real Cubic Number Fields

### Section 3. All totally real cubic fields L with discriminant 200000 < d < 300000 and multiplicity m = 4

In this second extension of the series of quadruplets
of unramified cyclic cubic extensions of real quadratic fields,
a totally 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
discovered, resp. analyzed, by myself (Daniel C. Mayer) on January 28, resp. 30, 2006, [1].
The capitulation type turned out to be G.19: (4,3,2,1),
up to now only known for complex quadratic base fields.

Continuation

Counter n Discriminant d Regulators R and class numbers h as pairs (R, h) Capitulation type
17 206776 (42.7, 3) (49.5, 3) (74.9, 3) (120.4, 3) a.2: (0,2,0,0)
18 209765 (61.9, 3) (62.0, 3) (68.6, 3) (69.7, 3) a.2: (0,2,0,0)
19 213913 (18.8, 6) (19.3, 3) (54.0, 3) (117.7, 3) a.3: (0,4,0,0)
20 214028 (72.9, 3) (84.8, 3) (85.7, 3) (89.3, 3) a.2: (0,0,3,0)
21 214712 (39.2, 6) (65.4, 3) (73.9, 3) (107.1, 3) G.19: (4,3,2,1)
22 219461 (24.7, 6) (45.9, 3) (60.9, 3) (72.7, 3) a.2: (0,0,3,0)
23 220217 (11.9, 6) (31.1, 3) (34.9, 3) (149.9, 3) a.3: (0,0,0,3)
24 250748 (55.7, 6) (68.6, 3) (75.9, 3) (96.0, 3) a.3: (3,0,0,0)
25 252977 (9.7, 9) (24.8, 3) (39.3, 3) (154.1, 3) a.1: (0,0,0,0)
26 259653 (66.5, 3) (69.4, 3) (88.9, 3) (93.7, 3) a.3*: (0,1,0,0)
27 265245 (86.3, 3) (91.3, 3) (91.5, 3) (114.4, 3) a.3: (0,0,4,0)
28 275881 (11.6, 3) (23.8, 3) (35.5, 3) (148.8, 3) a.2: (0,0,3,0)
29 283673 (29.8, 3) (30.4, 3) (37.8, 3) (180.2, 3) a.3*: (0,0,1,0)
30 298849 (13.1, 3) (24.7, 3) (47.8, 3) (124.4, 3) a.3: (0,0,0,1)

 References: [1] Daniel C. Mayer, 3-Capitulation over Quadratic Fields with Discriminant |d| < 3*105 and 3-Class Group of Type (3,3), Univ. Graz, Computer Centre, 2006.