Defining polynomial
|
\(x^{10} + 80 x^{9} + 2575 x^{8} + 41920 x^{7} + 350810 x^{6} + 1298690 x^{5} + 1053790 x^{4} + 419780 x^{3} + 741365 x^{2} + 5317450 x + 16794084\)
|
Invariants
| Base field: | $\Q_{17}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $8$ |
| Discriminant root field: | $\Q_{17}(\sqrt{3})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 17 }) }$: | $2$ |
| This field is not Galois over $\Q_{17}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{17}(\sqrt{3})$, 17.5.4.1 |
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: |
\( x^{5} + 17 \)
$\ \in\Q_{17}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{4} + 5z^{3} + 10z^{2} + 10z + 5$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $F_5$ (as 10T4) |
| Inertia group: | Intransitive group isomorphic to $C_5$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $5$ |
| Wild slopes: | None |
| Galois mean slope: | $4/5$ |
| Galois splitting model: | Not computed |