The real quadratic base field K with discriminant d = 710652
710652 is the smallest discriminant of a real quadratic field K
with capitulation type b.10 (0,0,4,3),
symbolic order V4,4, and group G=Gal(K2|K) in CBF1b(6,8)
of lower than second maximal class (e = 4).
710652 is also the smallest discriminant of a real quadratic field K
for which 2 of the 4 unramified cyclic cubic extensions Ni|K
have non-galois absolute cubic subfields Li|Q with 3-class number 9.
We give the complete data needed to determine the capitulation type and the group G=Gal(K2|K).
Computed on November 09, 2009, at the University of Graz, Computer Centre [1,2].
| Counter, n = 94 | Discriminant, d = 710652 | 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 | |
| 473767 | 1124 | 1 | 13.8 | |
| The non-Galois absolute cubic subfields (L1,L2,L3,L4) of the four unramified cyclic cubic relative extensions N|K | ||||
| Regulators, R | 35.2 | 68.0 | 198.7 | 222.0 |
| Class numbers, h | 18 | 9 | 3 | 3 |
| Polynomials, p(X) = X3 - C*X - D, with d(p) = i2*d | ||||
| (C,D) | (330,2092) | (186,76) | (690,6908) | (807,8770) |
| Indices, i | 6 | 6 | 6 | 6 |
| Fundamental units, e1 = (U1 + V1*x + W1*x2)/T1, with P(x) = 0 | ||||
| U1 | 13 | 77 | -13889 | 473 |
| V1 | 1 | 194 | -538 | 62 |
| W1 | 0 | 14 | 33 | 2 |
| T1 | 1 | 3 | 1 | 3 |
| Fundamental units, e2 = (U2 + V2*x + W2*x2)/T2, with P(x) = 0 | ||||
| U2 | 31 | 23 | 1034201 | 1777628 |
| V2 | 4 | 42 | 137434 | 660107 |
| W2 | 0 | 3 | 4536 | 35471 |
| T2 | 1 | 1 | 1 | 6 |
| Splitting primes, q | 31,577 | 103,109 | 73 | 37 |
| Associated quadratic forms, F = a*X2 + b*X*Y + c*Y2 | ||||
| (a,b,c) | (577,378,-246) | (109,772,-263) | (73,786,-318) | (37,794,-542) |
| Represented primes, q | 577 | 109 | 73 | 37 |
| Associated ideal cubes, (x + y*d1/2)/2, with 4*q3 = x2 - d*y2 | ||||
| (x,y) | (94414,112) | (59054,70) | (31216,37) | (5078,6) |
| Principalization | 0 | 0 | 4 | 3 |
| Capitulation type b.10: (0,0,4,3) | Group G in CBF1b(6,8) | Contents | ||