Defining polynomial
|
$( x^{2} + 16 x + 3 )^{6} + 17$
|
Invariants
| Base field: | $\Q_{17}$ |
|
| Degree $d$: | $12$ |
|
| Ramification index $e$: | $6$ |
|
| Residue field degree $f$: | $2$ |
|
| Discriminant exponent $c$: | $10$ |
|
| Discriminant root field: | $\Q_{17}$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{17})$ $=$ $\Gal(K/\Q_{17})$: | $D_6$ | |
| This field is Galois over $\Q_{17}.$ | ||
| Visible Artin slopes: | $[\ ]$ | |
| Visible Swan slopes: | $[\ ]$ | |
| Means: | $\langle\ \rangle$ | |
| Rams: | $(\ )$ | |
| Jump set: | undefined | |
| Roots of unity: | $288 = (17^{ 2 } - 1)$ |
|
Intermediate fields
| $\Q_{17}(\sqrt{3})$, $\Q_{17}(\sqrt{17})$, $\Q_{17}(\sqrt{17\cdot 3})$, 17.1.3.2a1.1 x3, 17.2.2.2a1.2, 17.2.3.4a1.2, 17.1.6.5a1.1 x3, 17.1.6.5a1.2 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical 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: |
\( x^{6} + 17 \)
$\ \in\Q_{17}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^5 + 6 z^4 + 15 z^3 + 3 z^2 + 15 z + 6$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $12$ |
| Galois group: | $D_6$ (as 12T3) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $6$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.8333333333333334$ |
| Galois splitting model: | not computed |