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 |