 # The real quadratic base field K with discriminant d = 540365

540365 is the smallest discriminant of a real quadratic field K
with capitulation type c.21 (0,2,3,1),
symbolic order X3, and non-terminal group G=Gal(K2|K) in CBF2a(5,6).

Important remark: This example shows that the strict leaf conjecture
that only terminal nodes (leaves) can occur as Gal(K2|K) is false.
However, it might still be true that a group cannot occur as Gal(K2|K),
if one of its successors is a leaf in CBFb with the same capitulation type.

We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed on January 01, 2008, at the University of Manitoba's Computer Centre, Winnipeg City [1,2].

Counter, n = 66 Discriminant, d = 540365 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
15437 21 2 9.6
The non-Galois absolute cubic subfields (L1,L2,L3,L4) of the four unramified cyclic cubic relative extensions N|K
Regulators, R 39.1 116.3 118.9 131.6
Class numbers, h 9 3 3 3
Polynomials, p(X) = X3 - C*X - D, with d(p) = i2*d
(C,D) (122,499) (68,163) (768,7247) (642,4961)
Indices, i 1 1 27 27
Fundamental units, e1 = (U1 + V1*x + W1*x2)/T1, with P(x) = 0
U1 16 -1487 238 -10294
V1 3 -394 32 -812
W1 0 60 1 41
T1 1 1 9 9
Fundamental units, e2 = (U2 + V2*x + W2*x2)/T2, with P(x) = 0
U2 83 -1936 -2870111 462048
V2 26 -531 -164494 75971
W2 2 80 7840 2660
T2 1 1 9 3
Splitting primes, q 31 97,181 19 67,229
Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2
(a,b,c) (31,715,-235) (-181,651,161) (19,713,-421) (-229,487,331)
Represented primes, q 31 -181 19 -229
Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2
(x,y) (649,1) (1661,7) (5143,7) (241747,329)
Principalization 0 2 3 1
Capitulation type c.21: (0,2,3,1) Group G in CBF2a(5,6) 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.