Defining polynomial
\(x^{15} + 5 x^{13} + 70 x^{12} + 10 x^{11} + 331 x^{10} + 1970 x^{9} + 165 x^{8} - 15535 x^{7} + 28060 x^{6} + 10318 x^{5} + 504940 x^{4} + 183500 x^{3} + 248865 x^{2} - 505940 x + 539184\) |
Invariants
Base field: | $\Q_{17}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $5$ |
Residue field degree $f$: | $3$ |
Discriminant exponent $c$: | $12$ |
Discriminant root field: | $\Q_{17}(\sqrt{3})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 17 }) }$: | $3$ |
This field is not Galois over $\Q_{17}.$ | |
Visible slopes: | None |
Intermediate fields
17.3.0.1, 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: | 17.3.0.1 $\cong \Q_{17}(t)$ where $t$ is a root of \( x^{3} + x + 14 \) |
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: | $4$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_3\times F_5$ (as 15T8) |
Inertia group: | Intransitive group isomorphic to $C_5$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $12$ |
Tame degree: | $5$ |
Wild slopes: | None |
Galois mean slope: | $4/5$ |
Galois splitting model: | Not computed |