Defining polynomial
\(x^{15} + 13 x^{12} + 3 x^{10} + 13 x^{9} - 174 x^{7} - 45 x^{6} + 3 x^{5} + 513 x^{4} + 63 x^{2} - 216 x + 109\) |
Invariants
Base field: | $\Q_{2}$ |
Degree $d$: | $15$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $5$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 2 }) }$: | $5$ |
This field is not Galois over $\Q_{2}.$ | |
Visible slopes: | None |
Intermediate fields
2.3.2.1, 2.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 2.5.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{5} + x^{2} + 1 \) |
Relative Eisenstein polynomial: | \( x^{3} + 2 \) $\ \in\Q_{2}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + z + 1$ |
Associated inertia: | $2$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_5\times S_3$ (as 15T4) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $10$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | Not computed |