Defining polynomial
|
\(x^{18} - 68 x^{15} + 6936 x^{12} - 225998 x^{9} + 9354352 x^{6} - 72412707 x^{3} + 217238121\)
|
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $18$ |
| Ramification exponent $e$: | $3$ |
| Residue field degree $f$: | $6$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{17}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 17 }) }$: | $9$ |
| This field is not Galois over $\Q_{17}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{17}(\sqrt{3})$, 17.3.0.1, 17.6.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 17.6.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of
\( x^{6} + 2 x^{4} + 10 x^{2} + 3 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{3} + 17 t^{2} \)
$\ \in\Q_{17}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{2} + 3z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $S_3\times C_9$ (as 18T16) |
| Inertia group: | Intransitive group isomorphic to $C_3$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $18$ |
| Tame degree: | $3$ |
| Wild slopes: | None |
| Galois mean slope: | $2/3$ |
| Galois splitting model: | Not computed |