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) |