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

## 213. Complex Quadratic Field K of Type D.16 (142;4) with Discriminant d = -311316

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 July 18, 2011, at the Computer Centre of the University Graz.

Counter, n = 213 Discriminant, d = -311316 Class group of type (36,6) Class number, h = 216 Conductor, f = 1
The 4 unramified cyclic cubic relative extensions (N1,N2,N3,N4) of K: ε = 4, η = 2
3-class groups, Cl3(Ni) (27,9,3) (9,9,9) (9,9,9) (3,3,3,3)
The 4 non-Galois absolute cubic subfields (L1,L2,L3,L4) of the Ni: triple excited state ↑↑↑
Regulators, R 19.26 32.31 39.76 59.34
3-class numbers, h3 9 9 9 3
Polynomials, p(X) = X3 - B*X2 + C*X - D, with d(p) = i2*d
(B,C,D) (8,10,204) (17,304,84) (18,129,22) (20,137,106)
Indices, i 2 16 3 2
Fundamental units, e = (U + V*z + W*z2)/T, with p(z) = 0
U -6643 -1886090 14061169 1070811344183
V -1839 6649037 -83103035 -1336000634855
W 275 254609 15131977 138758305357
T 1 16 3 1
Splitting primes, q 2797 1069 1153 313,661
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) AB=(145,-128,565) A4B2=(141,-24,553) A4=(166,-10,469) Bicyclic
Counter 2 3 1 4
Represented primes, q 2797 1069 1153 313,661
4 quadratic forms, F = a*X2 + b*X*Y + c*Y2, of order 3, possible generators of transfer kernels
(a,b,c) B=(70,38,1117) A3B=(301,-256,313) A3B2=(118,26,661) A3=(181,-2,430)
Counter 1 2 3 4
Represented primes, q 1117 313 661 181
4 associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2
(x,y) (74254,14) (8822,12) (10390,58) (3992,5)
Transfer kernels 4 2 1 4
Punctured transfer kernel type D.16 ((1,4,2);4) Cl3(N-4) = (9,9,9,9,3,3), w = 10 Navigation

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