 # The real quadratic base field K with discriminant d = 494236

494236 is the smallest discriminant of a real quadratic field K
with an excited state of capitulation type a.3 (3,0,0,0),
symbolic order Y4, and group G=Gal(K2|K) in CF2a(6) of maximal class.

We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed on February 04, 2008, at the University of Graz, Computer Centre [1,2].

Counter, n = 58 Discriminant, d = 494236 3-class group of type (3,3) 3-class number, h = 9 Conductor, f = 1
Fundamental unit, e0 = (U0 + V0*x)/T0, with x2 = d, and regulator, R
U0 V0 T0 R
4872576100076 13861857765 1 29.9
The non-Galois absolute cubic subfields (L1,L2,L3,L4) of the four unramified cyclic cubic relative extensions N|K
Regulators, R 36.4 60.1 84.8 178.4
Class numbers, h 9 3 3 3
Polynomials, p(X) = X3 - C*X - D, with d(p) = i2*d
(C,D) (547,4114) (82,92) (187,268) (154,684)
Indices, i 20 2 7 2
Fundamental units, e1 = (U1 + V1*x + W1*x2)/T1, with P(x) = 0
U1 162 -7 -23 9767
V1 27 -5 24 2886
W1 1 1 2 203
T1 20 1 7 1
Fundamental units, e2 = (U2 + V2*x + W2*x2)/T2, with P(x) = 0
U2 -7056 -67 2409 -63521
V2 -131 7 1849 -2889
W2 15 0 129 513
T2 2 1 1 1
Splitting primes, q 37 139 307 181,523
Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2
(a,b,c) (10,686,-591) (30,674,-333) (14,678,-617) (3,698,-586)
Represented primes, q 37 139,173 307,197 523,113
Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2
(x,y) (23204,33) (210252,299) (5706,2) (11502,16)
Principalization 3 0 0 0
Capitulation type a.3: (3,0,0,0) Group G in CF2a(6) Contents

 References:  Daniel C. Mayer, Two-Stage Towers of 3-Class Fields over Quadratic Fields, (Latest Update) Univ. Graz, 2008.  Daniel C. Mayer, 3-Capitulation over Quadratic Fields with Discriminant |d| < 106 and 3-Class Group of Type (3,3), (Latest Update) Univ. Graz, Computer Centre, 2008.