Defining polynomial
|
\(x^{8} + 180 x^{7} + 12170 x^{6} + 367200 x^{5} + 4222369 x^{4} + 1844460 x^{3} + 872480 x^{2} + 17027100 x + 190831059\)
|
Invariants
| Base field: | $\Q_{47}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{47}$ |
| Root number: | $1$ |
| $\card{ \Gal(K/\Q_{ 47 }) }$: | $8$ |
| This field is Galois over $\Q_{47}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{47}(\sqrt{5})$, $\Q_{47}(\sqrt{47})$, $\Q_{47}(\sqrt{47\cdot 5})$, 47.4.2.1, 47.4.3.1 x2, 47.4.3.2 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{47}(\sqrt{5})$ $\cong \Q_{47}(t)$ where $t$ is a root of
\( x^{2} + 45 x + 5 \)
|
| Relative Eisenstein polynomial: |
\( x^{4} + 47 \)
$\ \in\Q_{47}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z^{3} + 4z^{2} + 6z + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $D_4$ (as 8T4) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | Not computed |