Defining polynomial
|
\(x^{12} - 3604 x^{6} - 719321\)
|
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $10$ |
| Discriminant root field: | $\Q_{17}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 17 }) }$: | $12$ |
| This field is Galois over $\Q_{17}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{17}(\sqrt{3})$, 17.3.2.1 x3, 17.4.2.2, 17.6.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{17}(\sqrt{3})$ $\cong \Q_{17}(t)$ where $t$ is a root of
\( x^{2} + 16 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{6} + 255 t + 238 \)
$\ \in\Q_{17}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{5} + 6z^{4} + 15z^{3} + 3z^{2} + 15z + 6$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_3:C_4$ (as 12T5) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | None |
| Galois mean slope: | $5/6$ |
| Galois splitting model: | Not computed |