Defining polynomial
|
\(x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11\)
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $9$ |
| Ramification index $e$: | $1$ |
| Residue field degree $f$: | $9$ |
| Discriminant exponent $c$: | $0$ |
| Discriminant root field: | $\Q_{13}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{13})$ $=$$\Gal(K/\Q_{13})$: | $C_9$ |
| This field is Galois and abelian over $\Q_{13}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $10604499372 = (13^{ 9 } - 1)$ |
Intermediate fields
| 13.3.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 13.9.1.0a1.1 $\cong \Q_{13}(t)$ where $t$ is a root of
\( x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11 \)
|
| Relative Eisenstein polynomial: |
\( x - 13 \)
$\ \in\Q_{13}(t)[x]$
|
Ramification polygon
The ramification polygon is trivial for unramified extensions.