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 R_{6,4}, and group G=Gal(K_{2}K) in CBF^{2a}(8,11) of lower than second maximal class.
We give the complete data needed to determine the capitulation type and the group G=Gal(K_{2}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 L_{4} multiprecision is needed.
Counter, n = 268  Discriminant, d = 159208  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  6.8  17.3  26.4  54.2 
Class numbers, h  27  9  6  3 
Polynomials, p(X) = X^{3} + C*X + D, with d(p) = i^{2}*d  
(C,D)  (363,8710)  (35,1078)  (85,58)  (1497,24586) 
Indices, i  108  14  4  135 
Fundamental units, e = (U + V*x + W*x^{2})/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*X^{2} + b*X*Y + c*Y^{2}  
(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*d^{1/2})/2, with 4*q^{3} = x^{2}  d*y^{2}  
(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 CBF^{2a}(8,11)  Contents 

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