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 | ||