Defining polynomial
|
\(x^{15} + 5 x^{13} + 5 x^{12} + 10 x^{11} + 26 x^{10} + 20 x^{9} - 145 x^{7} + 70 x^{6} + 73 x^{5} + 315 x^{4} - 105 x^{3} + 200 x^{2} + 5 x + 1\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $12$ |
| Discriminant root field: | $\Q_{2}(\sqrt{5})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 2 }) }$: | $3$ |
| This field is not Galois over $\Q_{2}.$ | |
| Visible slopes: | None |
Intermediate fields
| 2.3.0.1, 2.5.4.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.3.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{3} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{5} + 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{4} + z^{3} + 1$ |
| 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 |