The complex quadratic base field K with discriminant d = -159208
159208 is the smallest absolute discriminant of a complex quadratic field K
with an excited state of capitulation type F.13 (2,3,4,3),
symbolic order R6,4, and group G=Gal(K2|K) in CBF2a(8,11) of lower than second maximal class.
We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed on June 19, 2003, at the University of Graz, Computer Centre [1], contained in [2,3].
For the fundamental unit of the fourth cubic field L4 multiprecision is needed.
| Counter, n = 268 | Discriminant, d = -159208 | 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 | 6.8 | 17.3 | 26.4 | 54.2 |
| Class numbers, h | 27 | 9 | 6 | 3 |
| Polynomials, p(X) = X3 + C*X + D, with d(p) = i2*d | ||||
| (C,D) | (-363,8710) | (-35,1078) | (85,58) | (-1497,-24586) |
| Indices, i | 108 | 14 | 4 | 135 |
| Fundamental units, e = (U + V*x + W*x2)/T, with P(x) = 0 | ||||
| U | 404 | 6252 | 6199 | 9146683660405 |
| V | -11 | -305 | 4897 | 294685375375 |
| W | -1 | -75 | -6243 | -11003614691 |
| T | 18 | 2 | 1 | 9 |
| Splitting primes, q | 13,547 | 127,2473 | 97,499 | 19,157 |
| Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2 | ||||
| (a,b,c) | (74,52,547) | (142,-20,281) | (137,-64,298) | (113,110,379) |
| Represented primes, q | 547 | 281,2473 | 137,499 | 113 |
| Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2 | ||||
| (x,y) | (22398,31) | (7586,14) | (310,8) | (2266,2) |
| Principalization | 2 | 3 | 4 | 3 |
| Capitulation type F.13: (2,3,4,3) | Group G in CBF2a(8,11) | Contents | ||