 # The complex quadratic base field K with discriminant d = -262744

262744 is the smallest absolute discriminant of a complex quadratic field K
with an excited state of capitulation type E.14 (2,4,4,1),
symbolic order X6, and group G=Gal(K2|K) in CBF2a(8,9) of second maximal class.

We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed on December 22, 2005, at the University of Graz, Computer Centre [1,2].
For the fundamental unit of the fourth cubic field L4 multiprecision is needed.

Counter, n = 463 Discriminant, d = -262744 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 7.0 25.3 52.1 85.0
Class numbers, h 27 6 3 3
Polynomials, p(X) = X3 + C*X + D, with d(p) = i2*d
(C,D) (-38,1384) (726,14096) (-309,3386) (339,1150)
Indices, i 14 162 27 27
Fundamental units, e = (U + V*x + W*x2)/T, with P(x) = 0
U -163 2113883 -2076167135905 -1157340152492040743
V -1 213521 -2007639781 -573543750656360737
W 1 4802 4363479455 -67377004517986643
T 7 27 9 9
Splitting primes, q 1039 1009 1999 967
Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2
(a,b,c) (65,44,1018) (175,-34,377) (89,42,743) (145,-34,455)
Represented primes, q 1039 1009 89,1999 967
Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2
(x,y) (50874,85) (56790,58) (1330,2) (58974,23)
Principalization 2 4 4 1
Capitulation type E.14: (2,4,4,1) Group G in CBF2a(8,9) Contents

 References:  Daniel C. Mayer, Two-Stage Towers of 3-Class Fields over Quadratic Fields, (Latest Update) Univ. Graz, 2008.  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.