Defining polynomial
|
$( x^{3} + x + 28 )^{2} + 31$
|
Invariants
| Base field: | $\Q_{31}$ |
| Degree $d$: | $6$ |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $3$ |
| Discriminant root field: | $\Q_{31}(\sqrt{31\cdot 3})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{31})$ $=$ $\Gal(K/\Q_{31})$: | $C_6$ |
| This field is Galois and abelian over $\Q_{31}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $29790 = (31^{ 3 } - 1)$ |
Intermediate fields
| $\Q_{31}(\sqrt{31\cdot 3})$, 31.3.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 31.3.1.0a1.1 $\cong \Q_{31}(t)$ where $t$ is a root of
\( x^{3} + x + 28 \)
|
| 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 degree: | $6$ |
| Galois group: | $C_6$ (as 6T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.5$ |
| Galois splitting model: | $x^{6} - x^{5} + 19 x^{4} - 13 x^{3} + 189 x^{2} - 49 x + 791$ |