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
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Discriminant, d = -262744
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3-class group of type (3,3)
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3-class number, h = 9
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Conductor, f = 1
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The non-Galois absolute cubic subfields (L1,L2,L3,L4)
of the four unramified cyclic cubic relative extensions N|K
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Regulators, R
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7.0
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25.3
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52.1
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85.0
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Class numbers, h
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27
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6
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3
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3
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Polynomials, p(X) = X3 + C*X + D, with d(p) = i2*d
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(C,D)
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(-38,1384)
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(726,14096)
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(-309,3386)
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(339,1150)
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Indices, i
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14
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162
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27
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27
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Fundamental units, e = (U + V*x + W*x2)/T, with P(x) = 0
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U
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-163
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2113883
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-2076167135905
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-1157340152492040743
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V
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-1
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213521
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-2007639781
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-573543750656360737
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W
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1
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4802
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4363479455
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-67377004517986643
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T
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7
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27
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9
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9
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Splitting primes, q
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1039
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1009
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1999
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967
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Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2
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(a,b,c)
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(65,44,1018)
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(175,-34,377)
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(89,42,743)
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(145,-34,455)
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Represented primes, q
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1039
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1009
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89,1999
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967
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Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2
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(x,y)
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(50874,85)
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(56790,58)
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(1330,2)
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(58974,23)
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Principalization
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2
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4
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4
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1
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Capitulation type E.14: (2,4,4,1)
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Group G in CBF2a(8,9)
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Contents
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