# 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 [2],
since his investigations were based on class group computations of Wada [1]
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 [3], 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: [1] Hideo Wada, A table of ideal class groups of imaginary quadratic fields, Proc. Japan Acad. 46 (1970), 401-403. [2] James R. Brink, The class field tower for imaginary quadratic number fields of type (3,3), Dissertation, Ohio State Univ., 1984. [3] 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 [4] Daniel C. Mayer, Two-Stage Towers of 3-Class Fields over Quadratic Fields, (Latest Update) Univ. Graz, 2008. [5] 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.

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