# The real quadratic base field K with discriminant d = 710652

710652 is the smallest discriminant of a real quadratic field K
with capitulation type b.10 (0,0,4,3),
symbolic order V4,4, and group G=Gal(K2|K) in CBF1b(6,8)
of lower than second maximal class (e = 4).

710652 is also the smallest discriminant of a real quadratic field K for which 2 of the 4 unramified cyclic cubic extensions Ni|K
have non-galois absolute cubic subfields Li|Q with 3-class number 9.

We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed on November 09, 2009, at the University of Graz, Computer Centre [1,2].

Counter, n = 94 Discriminant, d = 710652 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
473767 1124 1 13.8
The non-Galois absolute cubic subfields (L1,L2,L3,L4) of the four unramified cyclic cubic relative extensions N|K
Regulators, R 35.2 68.0 198.7 222.0
Class numbers, h 18 9 3 3
Polynomials, p(X) = X3 - C*X - D, with d(p) = i2*d
(C,D) (330,2092) (186,76) (690,6908) (807,8770)
Indices, i 6 6 6 6
Fundamental units, e1 = (U1 + V1*x + W1*x2)/T1, with P(x) = 0
U1 13 77 -13889 473
V1 1 194 -538 62
W1 0 14 33 2
T1 1 3 1 3
Fundamental units, e2 = (U2 + V2*x + W2*x2)/T2, with P(x) = 0
U2 31 23 1034201 1777628
V2 4 42 137434 660107
W2 0 3 4536 35471
T2 1 1 1 6
Splitting primes, q 31,577 103,109 73 37
Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2
(a,b,c) (577,378,-246) (109,772,-263) (73,786,-318) (37,794,-542)
Represented primes, q 577 109 73 37
Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2
(x,y) (94414,112) (59054,70) (31216,37) (5078,6)
Principalization 0 0 4 3
Capitulation type b.10: (0,0,4,3) Group G in CBF1b(6,8) Contents

 References: [1] Daniel C. Mayer, Two-Stage Towers of 3-Class Fields over Quadratic Fields, (Latest Update) Univ. Graz, 2009. [2] 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, 2009.