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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