Defining polynomial
|
\(x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272\)
|
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{17}$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 17 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{17}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{17}(\sqrt{3})$, $\Q_{17}(\sqrt{17})$, $\Q_{17}(\sqrt{17\cdot 3})$, 17.4.0.1, 17.4.2.1, 17.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 17.4.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of
\( x^{4} + 7 x^{2} + 10 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 153 x + 17 \)
$\ \in\Q_{17}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_4$ (as 8T2) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | Not computed |