 | |||
| |||
| |||
|
| |||
| |||
| |||
| |||
| |||
|
| |||
| |||
| |||
|
Content
The tables contain information about the maximal unramified soluble extension KH for all the 3043 real quadratic number fields K with discriminant dK smaller then 10000. These data were calculated with pari/gp, KANT and GAP. The results depend on the Generalized Riemann Hypothesis (GRH). But in any case the given KH is a subfield of the maximal unramified soluble extension of K and the length of the Hilbert class field tower is greater or equal then H.
Format
The data is given as a list. Each entry corresponds to a number field and consists of a vector
[dK, (KH :Q), H, Poly, h, g], where
| dK | is the discriminant of the quadratic field, |
| H | is length of the Hilbert class field tower, |
| Poly | is a generating Polynomial for KH (absolute reduced up to degree 48), |
| h | is a vector of length 3 containing the pairs [hKi,Cl(Ki)] for 0 ≤ i < H (Cl(Ki) is given in elem. div. notation.), |
| g | is a vector of length 2, which contains the identification of Gal(KH/Q) as Small Group according to GAP in the first and a structure description of it in the second component. |
An example is: [5,2,0,X^2-5,[[1,[]],[1,[]],[1,[]]],[[2,1],"C2"]]
Components
[dK=1, (KH :Q)=2, H=3, Poly=4, h=5, g=6]
Use a comma to separate higher dimensions:
h=[[hK="5,1,1",Cl(K)="5,1,2"],[hK1="5,2,1",Cl(K1)="5,2,2"],[hK2="5,3,1",Cl(K2)="5,3,2"]]
g=[Id(Gal)="6,1",Description(Gal)="6,2"]