# Quadruplets of Totally Real Cubic Number Fields

### Section 2. All totally real cubic fields L with discriminant 100000 < d < 200000 and multiplicity m = 4

This is the first extension of the series of unramified quadruplets and has been
discovered by myself (Daniel C. Mayer) in August 1991 on an Amdahl 5870 at the University of Manitoba, Winnipeg City [1].
Since there appeared only the well-known types a.1, a.2, and a.3 again,
my conjecture in 1991 was that real quadratic fields will always show only these capitulation types.

d = 142097 is the smallest discriminant with capitulation type a.3*,
where the distinguished sextic field of discriminant d3
has an exceptional 3-class group of type (3,3,3).
The other three sextic fields have 3-class groups of type (3,3).
(For the usual cases of capitulation type a.3,
the distinguished sextic field has a 3-class group of type (9,3).)

Continuation

Counter n Discriminant d Regulators R and class numbers h as pairs (R, h) Capitulation type
6 103809 (18.3, 3) (21.9, 3) (32.4, 3) (112.4, 3) a.3: (0,0,0,3)
7 114889 (8.5, 3) (20.4, 3) (26.6, 3) (61.1, 3) a.3: (0,0,2,0)
8 130397 (19.3, 6) (40.4, 3) (45.5, 3) (52.4, 3) a.3: (0,0,0,3)
9 142097 (8.8, 6) (18.3, 3) (26.9, 3) (116.7, 3) a.3*: (4,0,0,0)
10 151141 (14.1, 3) (15.8, 6) (27.6, 3) (97.1, 3) a.3: (0,3,0,0)
11 152949 (18.7, 9) (48.6, 3) (59.6, 3) (80.6, 3) a.1: (0,0,0,0)
12 153949 (28.7, 3) (29.9, 3) (39.8, 3) (67.3, 3) a.2: (1,0,0,0)
13 172252 (47.4, 3) (49.0, 3) (55.7, 3) (125.4, 3) a.3: (0,1,0,0)
14 173944 (31.9, 3) (54.6, 3) (69.5, 3) (99.2, 3) a.3*: (0,4,0,0)
15 184137 (28.5, 3) (30.8, 3) (39.4, 3) (167.3, 3) a.3: (0,3,0,0)
16 189237 (61.7, 3) (70.6, 3) (71.2, 3) (73.8, 3) a.2: (1,0,0,0)

 References: [1] Daniel C. Mayer, List of discriminants dL<200000 of totally real cubic fields L, arranged according to their multiplicities m and conductors f, Dept. of Comp. Sci., Univ. of Manitoba, 1991.