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)
