124363 is the smallest absolute discriminant of a complex quadratic field K
with capitulation type F.7 (4,4,1,1),
symbolic order R_{4,4}, and group G=Gal(K_{2}K) in CBF^{1a}(6,9) of lower than second maximal class.
This is the complex quadratic base field K with the smallest absolute discriminant
having the mysterious unknown capitulation type F.7 (4,4,1,1).
It was impossible for Brink to find this field in his 1984 thesis [2],
since his investigations were based on class group computations of Wada [1]
for odd discriminants d > 24000 and even discriminants d > 96000.
We give the complete data needed to determine the capitulation type and the group G=Gal(K_{2}K).
Computed on May 31, 2003, at the University of Graz, Computer Centre [3], contained in [4,5].
For the fundamental unit of the fourth cubic field L_{4} multiprecision is needed.
Counter, n = 210  Discriminant, d = 124363  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  8.7  10.5  10.7  33.4 
Class numbers, h  12  9  9  3 
Polynomials, p(X) = X^{3} + C*X + D, with d(p) = i^{2}*d  
(C,D)  (276,493)  (120,1901)  (356,2691)  (94,659) 
Indices, i  27  27  11  11 
Fundamental units, e = (U + V*x + W*x^{2})/T, with P(x) = 0  
U  64  1486  8975  48178465 
V  38  29  334  2156606 
W  1  8  34  1271046 
T  9  9  11  11 
Splitting primes, q  43,67  31,163  97,823  19,157 
Associated quadratic forms, F = a*X^{2} + b*X*Y + c*Y^{2}  
(a,b,c)  (67,49,473)  (163,13,191)  (41,21,761)  (157,83,209) 
Represented primes, q  67  163  41,823  157 
Associated ideal cubes, (x + y*d^{1/2})/2, with 4*q^{3} = x^{2}  d*y^{2}  
(x,y)  (840,2)  (3351,7)  (389,1)  (657,11) 
Principalization  4  4  1  1 
Capitulation type F.7: (4,4,1,1)  Group G in CBF^{1a}(6,9)  Contents 

Navigation Center 
Back to Algebra 