Defining polynomial
|
\(x^{18} + 6 x^{16} + 84 x^{15} + 15 x^{14} + 420 x^{13} + 3011 x^{12} + 840 x^{11} + 11367 x^{10} + 20020 x^{9} + 18411 x^{8} + 186480 x^{7} + 1998610 x^{6} + 147588 x^{5} + 1446885 x^{4} - 3479756 x^{3} + 734952 x^{2} + 1984584 x + 9683028\)
|
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $18$ |
| Ramification exponent $e$: | $6$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{17}(\sqrt{17})$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 17 }) }$: | $6$ |
| This field is not Galois over $\Q_{17}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{17}(\sqrt{17})$, 17.3.0.1, 17.3.2.1, 17.6.3.1, 17.6.5.1, 17.9.6.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 17.3.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of
\( x^{3} + x + 14 \)
|
| Relative Eisenstein polynomial: |
\( x^{6} + 17 \)
$\ \in\Q_{17}(t)[x]$
|
Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_6\times S_3$ (as 18T6) |
| Inertia group: | Intransitive group isomorphic to $C_6$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $6$ |
| Wild slopes: | None |
| Galois mean slope: | $5/6$ |
| Galois splitting model: | Not computed |