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) |