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 | ||