Defining polynomial over unramified subextension
| $x^{4} + 7d_{0}$ |
Invariants
| Residue field characteristic: | $7$ |
| Degree: | $12$ |
| Base field: | $\Q_{7}$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $9$ |
| Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $2$ (complete) |
| Ambiguity: | $6$ |
| Mass: | $1$ |
| Absolute Mass: | $1/3$ |
Varying
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[2]$ |
| Jump Set: | undefined |
Galois groups and Hidden Artin slopes
Fields
Showing all 2
Download displayed columns for results| Label | Polynomial | Galois group | Galois degree | $\#\Aut(K/\Q_p)$ | Hidden Artin slopes | Ind. of Insep. | Assoc. Inertia | Jump Set |
|---|---|---|---|---|---|---|---|---|
| 7.3.4.9a1.1 | $( x^{3} + 6 x^{2} + 4 )^{4} + 7 x$ | $D_4 \times C_3$ (as 12T14) | $24$ | $6$ | $[\ ]^{2}$ | $[0]$ | $[2]$ | undefined |
| 7.3.4.9a1.2 | $( x^{3} + 6 x^{2} + 4 )^{4} + 7$ | $D_4 \times C_3$ (as 12T14) | $24$ | $6$ | $[\ ]^{2}$ | $[0]$ | $[2]$ | undefined |