Defining polynomial
|
\(x^{22} + x^{2} - x + 77\)
|
Invariants
| Base field: | $\Q_{193}$ |
| Degree $d$: | $22$ |
| Ramification exponent $e$: | $1$ |
| Residue field degree $f$: | $22$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{193}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 193 }) }$: | $22$ |
| This field is Galois and abelian over $\Q_{193}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{193}(\sqrt{5})$, 193.11.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 193.22.0.1 $\cong \Q_{193}(t)$ where $t$ is a root of
\( x^{22} + x^{2} - x + 77 \)
|
| Relative Eisenstein polynomial: |
\( x - 193 \)
$\ \in\Q_{193}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.
Invariants of the Galois closure
| Galois group: | $C_{22}$ (as 22T1) |
| Inertia group: | trivial |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $22$ |
| Tame degree: | $1$ |
| Wild slopes: | None |
| Galois mean slope: | $0$ |
| Galois splitting model: | Not computed |