 # The complex quadratic base field K with discriminant d = -21668

21668 is the smallest absolute discriminant of a complex quadratic field K
with an excited state of capitulation type H.4 (2,1,1,1),
symbolic order Z5, and group G=Gal(K2|K) in CBF2b(7,8) of second maximal class.

This is the complex quadratic base field K with the smallest absolute discriminant
having an excited state of a capitulation type in general.
Our attention has been drawn to this field in November, 1989,
since one of the four associated absolute cubic fields has class number 9,
whereas the smaller occurrences of capitulation type H.4, d = -3896 and d = -6583,
have all four associated absolute cubic fields with class number 3.
This inspired our conjecture about the impact of 3-class numbers on the structure of G=Gal(K2|K),
which is now proved in Theorem 2 of 2008 Year of Mathematics.

It should be mentioned that Brink has determined the capitulation type of this field
in his 1984 thesis  already. However, he did not investigate connections with 3-class numbers
and did not use the distinction between ground states and excited states.

We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed in November, 1989, at the University of Graz, Computer Centre , contained in , and analyzed in [4,5].

Counter, n = 17 Discriminant, d = -21668 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 4.5 4.6 11.7 27.3
Class numbers, h 6 9 3 3
Polynomials, p(X) = X3 + C*X + D, with d(p) = i2*d
(C,D) (62,-212) (5,-28) (-115,-526) (17,-282)
Indices, i 10 1 8 10
Fundamental units, e = (U + V*x + W*x2)/T, with P(x) = 0
U 3 5 3542 2123287
V -1 -2 231 -98309
W 0 0 -41 -48121
T 1 1 4 5
Splitting primes, q 43,769 7,1093 13,277 31,829
Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2
(a,b,c) (57,50,106) (58,42,101) (21,2,258) (41,12,133)
Represented primes, q 113,769 101,1093 277 41,829
Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2
(x,y) (2094,8) (1398,10) (9178,6) (504,1)
Principalization 2 1 1 1
Capitulation type H.4: (2,1,1,1) Group G in CBF2b(7,8) Contents

 References:  James R. Brink, The class field tower for imaginary quadratic number fields of type (3,3), Dissertation, Ohio State Univ., 1984.  Daniel C. Mayer, Dihedral fields of degree 2p, Univ. Graz, 1989.  Daniel C. Mayer, Principalization in complex S3-fields, Congressus Numerantium 80 (1991), 73 - 87  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.