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 X_{6}, and group G=Gal(K_{2}K) in CBF^{2a}(8,9) of second maximal class.
We give the complete data needed to determine the capitulation type and the group G=Gal(K_{2}K).
Computed on December 22, 2005, at the University of Graz, Computer Centre [1,2].
For the fundamental unit of the fourth cubic field L_{4} multiprecision is needed.
Counter, n = 463

Discriminant, d = 262744

3class group of type (3,3)

3class number, h = 9

Conductor, f = 1

The nonGalois absolute cubic subfields (L_{1},L_{2},L_{3},L_{4})
of the four unramified cyclic cubic relative extensions NK

Regulators, R

7.0

25.3

52.1

85.0

Class numbers, h

27

6

3

3

Polynomials, p(X) = X^{3} + C*X + D, with d(p) = i^{2}*d

(C,D)

(38,1384)

(726,14096)

(309,3386)

(339,1150)

Indices, i

14

162

27

27

Fundamental units, e = (U + V*x + W*x^{2})/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*X^{2} + b*X*Y + c*Y^{2}

(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*d^{1/2})/2, with 4*q^{3} = x^{2}  d*y^{2}

(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 CBF^{2a}(8,9)

Contents
