Defining polynomial over unramified subextension
| $x^{7} + 2$ |
Invariants
| Residue field characteristic: | $2$ |
| Degree: | $14$ |
| Base field: | $\Q_{2}$ |
| Ramification index $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $12$ |
| Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $1$ (complete) |
| Ambiguity: | $2$ |
| Mass: | $1$ |
| Absolute Mass: | $1/2$ |
Varying
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[3]$ |
| Jump Set: | $[7]$ |
Galois groups and Hidden Artin slopes
Fields
Showing all 1
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 |
|---|---|---|---|---|---|---|---|---|
| 2.2.7.12a1.1 | $( x^{2} + x + 1 )^{7} + 2$ | $(C_7:C_3) \times C_2$ (as 14T5) | $42$ | $2$ | $[\ ]^{3}$ | $[0]$ | $[3]$ | $[7]$ |