The complex quadratic base field K with discriminant d = -4027
This example has been investigated by
Scholz and Taussky [1],
Heider and Schmithals [2],
Brink [3],
and in our previous papers [4,5].
We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed in October, 1989, at the University of Graz, Computer Centre [4].
| Counter, n = 2 | Discriminant, d = -4027 | 3-class group of type (3,3) | 3-class number, h = 9 | Conductor, f = 1 |
|---|---|---|---|---|
| The non-Galois absolute cubic subfields (L1,L2,L3,L4) of the four unramified cyclic cubic relative extensions N|K | ||||
| Regulators, R | 2.3 | 2.5 | 4.70 | 4.75 |
| Class numbers, h | 6 | 6 | 3 | 3 |
| Polynomials, p(X) = X3 + C*X + D, with d(p) = i2*d | ||||
| (C,D) | (10,-1) | (-44,-113) | (43,-56) | (-8,-15) |
| Indices, i | 1 | 1 | 10 | 1 |
| Fundamental units, e = (U + V*x + W*x2)/T, with P(x) = 0 | ||||
| U | 0 | -28 | 18 | -7 |
| V | 1 | -4 | -13 | 2 |
| W | 0 | 1 | -1 | 0 |
| T | 1 | 1 | 10 | 1 |
| Splitting primes, q | 43 | 13 | 19 | 61 |
| Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2 | ||||
| (a,b,c) | (29,27,41) | (13,9,79) | (19,1,53) | (17,11,61) |
| Represented primes, q | 29, 43 | 13 | 19 | 17, 61 |
| Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2 | ||||
| (x,y) | (182,4) | (69,1) | (153,1) | (125,1) |
| Principalization | 2 | 3 | 3 | 1 |
| Capitulation type D.10: (2,3,3,1) | Group G in CBF1a(4,5) | Contents | ||
