The real quadratic base field K with discriminant d = 214712
214712 is the smallest discriminant of a real quadratic field K
with capitulation type G.19 (4,3,2,1),
symbolic order Z, and group G=Gal(K2|K) in CBF1b(5,6) of second maximal class.
Similarly as over complex quadratic fields,
in none of the four unramified cyclic cubic extensions N|K
the complete 3-class group of K becomes principal.
We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed on January 28, 2006, at the University of Graz, Computer Centre [1,2].
| Counter, n = 21 | Discriminant, d = 214712 | 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 | |
| 6315163023 | 27257524 | 1 | 23.3 | |
| The non-Galois absolute cubic subfields (L1,L2,L3,L4) of the four unramified cyclic cubic relative extensions N|K | ||||
| Regulators, R | 39.2 | 65.4 | 73.9 | 107.1 |
| Class numbers, h | 6 | 3 | 3 | 3 |
| Polynomials, p(X) = X3 - C*X - D, with d(p) = i2*d | ||||
| (C,D) | (83,230) | (993,358) | (1137,8534) | (74,168) |
| Indices, i | 2 | 135 | 135 | 2 |
| Fundamental units, e1 = (U1 + V1*x + W1*x2)/T1, with P(x) = 0 | ||||
| U1 | 19244 | 104 | 44655919 | 38613970 |
| V1 | 8819 | 248 | 7157054 | 21044036 |
| W1 | 859 | 8 | 193522 | 2199288 |
| T1 | 2 | 360 | 45 | 2 |
| Fundamental units, e2 = (U2 + V2*x + W2*x2)/T2, with P(x) = 0 | ||||
| U2 | -72 | -3762479 | -21503471 | -143 |
| V2 | -3 | -1288 | -3446386 | -84 |
| W2 | 1 | 3792 | -93188 | -9 |
| T2 | 2 | 15 | 45 | 1 |
| Splitting primes, q | 13 | 43 | 103 | 7 |
| Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2 | ||||
| (a,b,c) | (13,440,-406) | (-41,404,314) | (17,436,-362) | (7,454,-307) |
| Represented primes, q | -13 | -43 | -103 | -7 |
| Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2 | ||||
| (x,y) | (769658,1661) | (328530,709) | (2328898,5026) | (926,2) |
| Principalization | 4 | 3 | 2 | 1 |
| Capitulation type G.19: (4,3,2,1) | Group G in CBF1b(5,6) | Contents | ||