Defining polynomial
| $x^{3} + \pi$ |
Invariants
| Residue field characteristic: | $2$ |
| Degree: | $3$ |
| Base field: | 2.1.7.6a1.1 |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $2$ |
| Absolute Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $1$ (complete) |
| Ambiguity: | $1$ |
| Mass: | $1$ |
| Absolute Mass: | $1$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $C_{21}:C_6$ |
| Hidden Artin slopes: | $[\ ]^{6}$ |
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[6]$ |
| Jump Set: | $[21]$ |
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 |
|---|---|---|---|---|---|---|---|---|
| 2.1.21.20a1.1 | $x^{21} + 2$ | $C_{21}:C_6$ (as 21T11) | $126$ | $1$ | $[\ ]^{6}$ | $[0]$ | $[6]$ | $[21]$ |