Memorial 2009



Dedicated to the Memory of

Alexander Aigner

in the Year 2009:

Quadruplets of Totally Real Cubic Number Fields

Section 10. All totally real cubic fields L with discriminant 900000 < d < 106 and multiplicity m = 4

Breaking through beyond Ennola and Turunen's domain [1] and reaching the target

Two real quadratic fields with four unramified cyclic cubic extensions
of principal factorization type Alpha 1 were found in this tenth range of length 100000.
We discovered and analyzed them on December 17, 2009, [2].

Further, an eleventh and twelfth unexpected and surprising result occurred (green color).
Two real quadratic fields with four unramified cyclic cubic extensions
of principal factorization type Delta 1 have been
discovered and analyzed on December 17, 2009, [2].
The capitulation type turned out to be G.19: (4,3,2,1), resp. H.4: (4,4,4,1),
up to now only known for complex quadratic base fields.

d=945813 is a second discriminant where
type c.21 appears with non-terminal group G=Gal(K2|K) in CBF2a(5,6).
(Discovered and analyzed [2] on December 17, 2009.)

Summary

Counter n Discriminant d Regulators R and class numbers h as pairs (R, h) Capitulation type
127 902333 (101.5, 3) (108.8, 3) (118.9, 3) (140.2, 3) a.3: (0,0,0,1)
128 907629 (108.4, 3) (120.6, 3) (154.7, 3) (198.8, 3) a.2: (0,0,3,0)
129 907709 (49.3, 6) (87.9, 6) (98.7, 3) (120.6, 3) a.2: (0,0,3,0)
130 908241 (89.1, 3) (109.5, 3) (120.3, 3) (534.0, 3) a.3: (0,0,0,3)
131 916181 (116.9, 3) (119.2, 3) (120.3, 3) (189.9, 3) a.3*: (0,4,0,0)
132 935665 (23.4, 6) (36.7, 3) (38.8, 3) (633.1, 3) a.3: (0,0,1,0)
133 939569 (35.7, 3) (53.1, 3) (73.9, 3) (289.7, 3) a.3: (0,4,0,0)
134 940593 (32.3, 9) (61.2, 3) (74.8, 3) (406.5, 3) a.1: (0,0,0,0)
135 942961 (30.3, 3) (45.0, 12) (51.9, 3) (85.4, 3) a.3*: (3,0,0,0)
136 943077 (22.9, 21) (125.8, 3) (146.0, 3) (194.9, 3) G.19: (4,3,2,1)
137 944760 (249.2, 3) (280.3, 3) (290.7, 3) (298.7, 3) a.3: (0,0,4,0)
138 945813 (55.2, 9) (145.0, 3) (156.7, 3) (212.0, 3) c.21: (0,2,3,1)
139 957013 (41.5, 6) (68.1, 3) (108.7, 3) (209.8, 3) H.4: (2,1,2,2)
140 957484 (101.2, 3) (106.1, 3) (180.6, 3) (306.6, 3) a.2: (0,2,0,0)
141 959629 (69.9, 3) (76.2, 3) (125.1, 3) (203.7, 3) a.2: (0,0,0,4)
142 966053 (113.6, 3) (128.2, 3) (131.2, 3) (159.7, 3) a.3*: (4,0,0,0)
143 966489 (19.5, 9) (60.6, 3) (68.0, 3) (566.8, 3) a.1: (0,0,0,0)
144 967928 (180.5, 3) (198.1, 3) (198.5, 3) (211.1, 3) a.3*: (0,0,2,0)
145 974157 (143.5, 3) (143.8, 3) (157.2, 3) (208.6, 3) a.3*: (0,0,0,2)
146 980108 (96.4, 6) (170.1, 3) (175.0, 3) (213.2, 3) a.2: (0,0,0,4)
147 982049 (24.9, 6) (45.3, 3) (46.1, 6) (274.3, 3) a.3*: (0,0,2,0)
148 993349 (70.1, 3) (86.8, 3) (129.8, 3) (203.4, 3) a.2: (0,0,0,4)
149 994008 (103.8, 6) (185.0, 3) (252.9, 3) (298.0, 3) a.2: (1,0,0,0)


References:

[1] V. Ennola and R. Turunen,
On totally real cubic fields,
Math. Comp. 44 (1985), no. 170, 495-518.

[2] 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.

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