We give complete data needed to determine
the second 3-class group G=Gal(F_{3}^{2}(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 (N_{1},N_{2},N_{3},N_{4}) of K: ε = 4, η = 1 | ||||
3-class groups, Cl_{3}(N_{i}) | (9,9,9) | (27,9,3) | (9,3,3) | (9,3,3) |
The 4 non-Galois absolute cubic subfields (L_{1},L_{2},L_{3},L_{4}) of the N_{i}: double excited state ↑↑ | ||||
Regulators, R | 4.63 | 5.09 | 8.03 | 29.50 |
3-class numbers, h_{3} | 9 | 9 | 3 | 3 |
Tracefree polynomials, p(X) = X^{3} + C*X + D, with d(p) = i^{2}*d | ||||
(C,D) | (-282,-2387) | (4,57) | (22,-41) | (-66,-1555) |
Indices, i | 27 | 1 | 1 | 27 |
Fundamental units, e = (U + V*z + W*z^{2})/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*X^{2} + b*X*Y + c*Y^{2}, of order 9, expressed by A=(25,11,881), establishing the Artin map | ||||
(a,b,c) | A^{4}B^{2}=(119,9,185) | Bicyclic | AB=(35,-19,631) | A^{7}=(53,-1,415) |
Counter | 3 | 4 | 2 | 1 |
Represented primes, q | 313 | 487,673 | 631 | 2131 |
4 quadratic forms, F = a*X^{2} + b*X*Y + c*Y^{2}, of order 3, possible generators of transfer kernels | ||||
(a,b,c) | B=(35,9,629) | A^{3}B=(85,9,259) | A^{3}B^{2}=(145,-139,185) | A^{6}=(119,93,203) |
Counter | 1 | 2 | 3 | 4 |
Represented primes, q | 673 | 997 | 487 | 229 |
4 associated ideal cubes, (x + y*d^{1/2})/2, with 4*q^{3} = x^{2} - d*y^{2} | ||||
(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) | Cl_{3}(N^{-}_{4}) = (9,9,9,3,3), w = 8 | Navigation |
* |
Web master's e-mail address:
contact@algebra.at |
* |
Back to Algebra |