The complex quadratic base field K with discriminant d = -124363
124363 is the smallest absolute discriminant of a complex quadratic field K
with capitulation type F.7 (4,4,1,1),
symbolic order R4,4, and group G=Gal(K2|K) in CBF1a(6,9) of lower than second maximal class.
This is the complex quadratic base field K with the smallest absolute discriminant
having the mysterious unknown capitulation type F.7 (4,4,1,1).
It was impossible for Brink to find this field in his 1984 thesis [2],
since his investigations were based on class group computations of Wada [1]
for odd discriminants d > -24000 and even discriminants d > -96000.
We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed on May 31, 2003, at the University of Graz, Computer Centre [3], contained in [4,5].
For the fundamental unit of the fourth cubic field L4 multiprecision is needed.
| Counter, n = 210 | Discriminant, d = -124363 | 3-class group of type (3,3) | 3-class number, h = 9 | Conductor, f = 1 |
|---|---|---|---|---|
| The non-Galois absolute cubic subfields (L1,L2,L3,L4) of the four unramified cyclic cubic relative extensions N|K | ||||
| Regulators, R | 8.7 | 10.5 | 10.7 | 33.4 |
| Class numbers, h | 12 | 9 | 9 | 3 |
| Polynomials, p(X) = X3 + C*X + D, with d(p) = i2*d | ||||
| (C,D) | (276,-493) | (-120,-1901) | (-356,-2691) | (94,659) |
| Indices, i | 27 | 27 | 11 | 11 |
| Fundamental units, e = (U + V*x + W*x2)/T, with P(x) = 0 | ||||
| U | 64 | 1486 | 8975 | -48178465 |
| V | -38 | 29 | 334 | -2156606 |
| W | 1 | -8 | -34 | 1271046 |
| T | 9 | 9 | 11 | 11 |
| Splitting primes, q | 43,67 | 31,163 | 97,823 | 19,157 |
| Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2 | ||||
| (a,b,c) | (67,49,473) | (163,-13,191) | (41,21,761) | (157,-83,209) |
| Represented primes, q | 67 | 163 | 41,823 | 157 |
| Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2 | ||||
| (x,y) | (840,2) | (3351,7) | (389,1) | (657,11) |
| Principalization | 4 | 4 | 1 | 1 |
| Capitulation type F.7: (4,4,1,1) | Group G in CBF1a(6,9) | Contents | ||