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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