The complex quadratic base field K with discriminant d = -262744
262744 is the smallest absolute discriminant of a complex quadratic field K
with an excited state of capitulation type E.14 (2,4,4,1),
symbolic order X6, and group G=Gal(K2|K) in CBF2a(8,9) of second maximal class.
We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed on December 22, 2005, at the University of Graz, Computer Centre [1,2].
For the fundamental unit of the fourth cubic field L4 multiprecision is needed.
| Counter, n = 463 | Discriminant, d = -262744 | 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 | 7.0 | 25.3 | 52.1 | 85.0 |
| Class numbers, h | 27 | 6 | 3 | 3 |
| Polynomials, p(X) = X3 + C*X + D, with d(p) = i2*d | ||||
| (C,D) | (-38,1384) | (726,14096) | (-309,3386) | (339,1150) |
| Indices, i | 14 | 162 | 27 | 27 |
| Fundamental units, e = (U + V*x + W*x2)/T, with P(x) = 0 | ||||
| U | -163 | 2113883 | -2076167135905 | -1157340152492040743 |
| V | -1 | 213521 | -2007639781 | -573543750656360737 |
| W | 1 | 4802 | 4363479455 | -67377004517986643 |
| T | 7 | 27 | 9 | 9 |
| Splitting primes, q | 1039 | 1009 | 1999 | 967 |
| Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2 | ||||
| (a,b,c) | (65,44,1018) | (175,-34,377) | (89,42,743) | (145,-34,455) |
| Represented primes, q | 1039 | 1009 | 89,1999 | 967 |
| Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2 | ||||
| (x,y) | (50874,85) | (56790,58) | (1330,2) | (58974,23) |
| Principalization | 2 | 4 | 4 | 1 |
| Capitulation type E.14: (2,4,4,1) | Group G in CBF2a(8,9) | Contents | ||