Defining polynomial
\(x^{12} - 34 x^{8} + 289 x^{4} + 962948\) |
Invariants
Base field: | $\Q_{17}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $3$ |
Discriminant exponent $c$: | $9$ |
Discriminant root field: | $\Q_{17}(\sqrt{17})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 17 }) }$: | $12$ |
This field is Galois and abelian over $\Q_{17}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{17}(\sqrt{17})$, 17.3.0.1, 17.4.3.2, 17.6.3.1 |
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: | \( x^{4} + 17 t^{2} \) $\ \in\Q_{17}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{3} + 4z^{2} + 6z + 4$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_{12}$ (as 12T1) |
Inertia group: | Intransitive group isomorphic to $C_4$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $3$ |
Tame degree: | $4$ |
Wild slopes: | None |
Galois mean slope: | $3/4$ |
Galois splitting model: | Not computed |