Defining polynomial
|
\(x^{9} - 52 x^{6} + 676 x^{3} + 265837\)
|
Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $9$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{13}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 13 }) }$: | $9$ |
| This field is Galois and abelian over $\Q_{13}.$ | |
| Visible slopes: | None |
Intermediate fields
| 13.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 13.3.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of
\( x^{3} + 2 x + 11 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + 13 t^{2} \)
$\ \in\Q_{13}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{2} + 3z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_9$ (as 9T1) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $3$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | $x^{9} - x^{8} - 84 x^{7} + 121 x^{6} + 1940 x^{5} - 1706 x^{4} - 17861 x^{3} + 3544 x^{2} + 58012 x + 23977$ |