 # The complex quadratic base field K with discriminant d = -124363

124363 is the smallest absolute discriminant of a complex quadratic field K
with capitulation type F.7 (4,4,1,1),
symbolic order R4,4, and group G=Gal(K2|K) in CBF1a(6,9) of lower than second maximal class.

This is the complex quadratic base field K with the smallest absolute discriminant
having the mysterious unknown capitulation type F.7 (4,4,1,1).
It was impossible for Brink to find this field in his 1984 thesis ,
since his investigations were based on class group computations of Wada 
for odd discriminants d > -24000 and even discriminants d > -96000.

We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed on May 31, 2003, at the University of Graz, Computer Centre , contained in [4,5].
For the fundamental unit of the fourth cubic field L4 multiprecision is needed.

Counter, n = 210 Discriminant, d = -124363 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 8.7 10.5 10.7 33.4
Class numbers, h 12 9 9 3
Polynomials, p(X) = X3 + C*X + D, with d(p) = i2*d
(C,D) (276,-493) (-120,-1901) (-356,-2691) (94,659)
Indices, i 27 27 11 11
Fundamental units, e = (U + V*x + W*x2)/T, with P(x) = 0
U 64 1486 8975 -48178465
V -38 29 334 -2156606
W 1 -8 -34 1271046
T 9 9 11 11
Splitting primes, q 43,67 31,163 97,823 19,157
Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2
(a,b,c) (67,49,473) (163,-13,191) (41,21,761) (157,-83,209)
Represented primes, q 67 163 41,823 157
Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2
(x,y) (840,2) (3351,7) (389,1) (657,11)
Principalization 4 4 1 1
Capitulation type F.7: (4,4,1,1) Group G in CBF1a(6,9) Contents

 References:  Hideo Wada, A table of ideal class groups of imaginary quadratic fields, Proc. Japan Acad. 46 (1970), 401-403.  James R. Brink, The class field tower for imaginary quadratic number fields of type (3,3), Dissertation, Ohio State Univ., 1984.  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  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.