# Quadruplets of Totally Real Cubic Number Fields

### Section 4. All totally real cubic fields L with discriminant 300000 < d < 400000 and multiplicity m = 4

As usual, a single real quadratic field with four unramified cyclic cubic extensions
of principal factorization type Alpha 1 was found in this fourth range of length 100000.
We discovered it on April 26, 2006, and verified it on August 13, 2007, [1].

Further, a second unexpected and surprising result occurred in this range (green color).
A single real quadratic field with four unramified cyclic cubic extensions
of principal factorization type Delta 1 has been
discovered on April 28, 2006, and analyzed on June 13, 2006, [1].
The capitulation type turned out to be E.9: (4,1,3,4),
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
31 320785 (18.5, 3) (25.5, 3) (32.0, 6) (164.5, 3) a.3*: (0,1,0,0)
32 321053 (67.7, 3) (69.4, 3) (71.7, 3) (76.6, 3) a.3*: (0,4,0,0)
33 326945 (35.6, 3) (37.6, 3) (48.0, 3) (212.3, 3) a.3*: (0,1,0,0)
34 333656 (14.5, 15) (88.7, 3) (101.8, 3) (124.8, 3) a.3: (0,0,0,3)
35 335229 (70.9, 3) (72.4, 3) (89.3, 3) (127.8, 3) a.3*: (3,0,0,0)
36 341724 (106.5, 3) (112.8, 3) (138.6, 3) (180.4, 3) a.3: (0,0,0,2)
37 342664 (23.1, 9) (59.4, 3) (60.2, 3) (170.4, 3) E.9: (4,1,3,4)
38 358285 (15.8, 9) (57.0, 3) (69.1, 3) (139.0, 3) a.1: (0,0,0,0)
39 363397 (45.5, 3) (48.7, 3) (56.3, 3) (134.9, 3) a.3: (0,0,2,0)
40 371965 (58.4, 3) (63.9, 3) (71.4, 3) (162.4, 3) a.3: (0,0,0,3)
41 390876 (127.8, 3) (127.9, 3) (128.9, 3) (205.8, 3) a.2: (1,0,0,0)

 References: [1] 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.