# The complex quadratic base field K with discriminant d = -159208

159208 is the smallest absolute discriminant of a complex quadratic field K
with an excited state of capitulation type F.13 (2,3,4,3),
symbolic order R6,4, and group G=Gal(K2|K) in CBF2a(8,11) of lower than second maximal class.

We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed on June 19, 2003, at the University of Graz, Computer Centre [1], contained in [2,3].
For the fundamental unit of the fourth cubic field L4 multiprecision is needed.

Counter, n = 268 Discriminant, d = -159208 3-class group of type (3,3) 3-class number, h = 9 Conductor, f = 1
The non-Galois absolute cubic subfields (L1,L2,L3,L4) of the four unramified cyclic cubic relative extensions N|K
Regulators, R 6.8 17.3 26.4 54.2
Class numbers, h 27 9 6 3
Polynomials, p(X) = X3 + C*X + D, with d(p) = i2*d
(C,D) (-363,8710) (-35,1078) (85,58) (-1497,-24586)
Indices, i 108 14 4 135
Fundamental units, e = (U + V*x + W*x2)/T, with P(x) = 0
U 404 6252 6199 9146683660405
V -11 -305 4897 294685375375
W -1 -75 -6243 -11003614691
T 18 2 1 9
Splitting primes, q 13,547 127,2473 97,499 19,157
Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2
(a,b,c) (74,52,547) (142,-20,281) (137,-64,298) (113,110,379)
Represented primes, q 547 281,2473 137,499 113
Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2
(x,y) (22398,31) (7586,14) (310,8) (2266,2)
Principalization 2 3 4 3
Capitulation type F.13: (2,3,4,3) Group G in CBF2a(8,11) Contents

 References: [1] Daniel C. Mayer, Principalization in Unramified Cyclic Cubic Extensions of selected Quadratic Fields with Discriminant -200000 < d < -50000 and 3-Class Group of Type (3,3), Univ. Graz, Computer Centre, 2004 [2] Daniel C. Mayer, Two-Stage Towers of 3-Class Fields over Quadratic Fields, (Latest Update) Univ. Graz, 2008. [3] 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, 2008.