Invariants
| Residue field characteristic: | $2$ |
| Degree: | $7$ |
| Base field: | $\Q_{2}(\sqrt{2})$ |
| Ramification index $e$: | $1$ |
| Residue field degree $f$: | $7$ |
| Discriminant exponent $c$: | $0$ |
| Absolute Artin slopes: | $[3]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $1$ (complete) |
| Ambiguity: | $7$ |
| Mass: | $1$ |
| Absolute Mass: | $1/14$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $C_{14}$ |
| Hidden Artin slopes: | $[\ ]$ |
| Indices of inseparability: | $[2,0]$ |
| Associated inertia: | $[1]$ |
| Jump Set: | $[1,3]$ |
Fields
Showing all 1
Download displayed columns for results| Label | Polynomial $/ \Q_p$ | Galois group $/ \Q_p$ | Galois degree $/ \Q_p$ | $\#\Aut(K/\Q_p)$ | Hidden Artin slopes $/ \Q_p$ | Ind. of Insep. $/ \Q_p$ | Assoc. Inertia $/ \Q_p$ | Jump Set |
|---|---|---|---|---|---|---|---|---|
| 2.7.2.21a1.21 | $( x^{7} + x + 1 )^{2} + 4 ( x^{7} + x + 1 ) + 2$ | $C_{14}$ (as 14T1) | $14$ | $14$ | $[\ ]$ | $[2, 0]$ | $[1]$ | $[1, 3]$ |