Defining polynomial
|
\(x^{8} + 130 x^{6} + 32 x^{5} + 5967 x^{4} - 1888 x^{3} + 113114 x^{2} - 89184 x + 766577\)
|
Invariants
| Base field: | $\Q_{31}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $4$ |
| Discriminant root field: | $\Q_{31}$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 31 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{31}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{31}(\sqrt{3})$, $\Q_{31}(\sqrt{31})$, $\Q_{31}(\sqrt{31\cdot 3})$, 31.4.0.1, 31.4.2.1, 31.4.2.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 31.4.0.1 $\cong \Q_{31}(t)$ where $t$ is a root of
\( x^{4} + 3 x^{2} + 16 x + 3 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 31 \)
$\ \in\Q_{31}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 2$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_2\times C_4$ (as 8T2) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $4$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | Not computed |