# Quadratic Number Fields with 3-Class Group of Type (9,3)

## 42. Complex Quadratic Field K of Type D.17 (442;3) with Discriminant d = -87979

We give complete data needed to determine the second 3-class group G=Gal(F32(K)|K) and its punctured transfer kernel type (TKT).
Computed on August 10 and December 23, 2009, recomputed on December 06, 2010, at the Computer Centre of the University Graz.

Counter, n = 42 Discriminant, d = -87979 Class group of type (18,3) Class number, h = 54 Conductor, f = 1
The 4 unramified cyclic cubic relative extensions (N1,N2,N3,N4) of K: ε = 4, η = 1
3-class groups, Cl3(Ni) (9,9,9) (27,9,3) (9,3,3) (9,3,3)
The 4 non-Galois absolute cubic subfields (L1,L2,L3,L4) of the Ni: double excited state ↑↑
Regulators, R 4.63 5.09 8.03 29.50
3-class numbers, h3 9 9 3 3
Tracefree polynomials, p(X) = X3 + C*X + D, with d(p) = i2*d
(C,D) (-282,-2387) (4,57) (22,-41) (-66,-1555)
Indices, i 27 1 1 27
Fundamental units, e = (U + V*z + W*z2)/T, with p(z) = 0
U -20 7 -16 -4512968
V 1 2 8 -1200437
W 0 0 1 113992
T 3 1 1 9
Splitting primes, q 313 487,673 631 2131
3 quadratic forms, F = a*X2 + b*X*Y + c*Y2, of order 9, expressed by A=(25,11,881), establishing the Artin map
(a,b,c) A4B2=(119,9,185) Bicyclic AB=(35,-19,631) A7=(53,-1,415)
Counter 3 4 2 1
Represented primes, q 313 487,673 631 2131
4 quadratic forms, F = a*X2 + b*X*Y + c*Y2, of order 3, possible generators of transfer kernels
(a,b,c) B=(35,9,629) A3B=(85,9,259) A3B2=(145,-139,185) A6=(119,93,203)
Counter 1 2 3 4
Represented primes, q 673 997 487 229
4 associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2
(x,y) (34917,1) (13939,207) (17516,42) (4755,17)
Transfer kernels 2 3 4 4
Punctured transfer kernel type D.17 ((4,4,2);3) Cl3(N-4) = (9,9,9,3,3), w = 8 Navigation

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