The complex quadratic base field K with discriminant d = -50739
This example has neither been investigated by Brink [1]
nor in our previous paper [2].
We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed on July 31, 2007, at the University of Graz, Computer Centre [3,4].
| Counter, n = 78 | Discriminant, d = -50739 | 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 | 18.0 | 21.5 | 24.7 | 29.8 |
| Class numbers, h | 3 | 3 | 3 | 3 |
| Polynomials, p(X) = X3 + C*X + D, with d(p) = i2*d | ||||
| (C,D) | (48,-23) | (-204,-1129) | (-42,167) | (-186,-985) |
| Indices, i | 3 | 3 | 3 | 3 |
| Fundamental units, e = (U + V*x + W*x2)/T, with P(x) = 0 | ||||
| U | 134 | -345974 | -219811 | -75804979 |
| V | -518 | -18319 | 109904 | -4682296 |
| W | 497 | 2380 | 17330 | 602102 |
| T | 3 | 1 | 3 | 3 |
| Splitting primes, q | 127 | 43 | 823 | 61 |
| Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2 | ||||
| (a,b,c) | (105,-51,127) | (43,1,295) | (55,31,235) | (61,47,217) |
| Represented primes, q | 127 | 43 | 823 | 61 |
| Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2 | ||||
| (x,y) | (2389,7) | (517,1) | (41623,99) | (310,4) |
| Principalization | 3 | 1 | 4 | 4 |
| Capitulation type D.10: (3,1,4,4) | Group G in CBF1a(4,5) | Contents | ||