Defining polynomial
| $x^{3} + \pi$ |
Invariants
| Residue field characteristic: | $2$ |
| Degree: | $3$ |
| Base field: | $\Q_{2}(\sqrt{-1})$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $2$ |
| Absolute Artin slopes: | $[2]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $1$ (complete) |
| Ambiguity: | $1$ |
| Mass: | $1$ |
| Absolute Mass: | $1/2$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $D_{6}$ |
| Hidden Artin slopes: | $[\ ]^{2}$ |
| Indices of inseparability: | $[3,0]$ |
| Associated inertia: | $[2,1]$ |
| Jump Set: | $[3,6]$ |
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 |
|---|---|---|---|---|---|---|---|---|
| 2.1.6.8a1.1 | $x^{6} + 2 x^{3} + 2$ | $D_{6}$ (as 6T3) | $12$ | $2$ | $[\ ]^{2}$ | $[3, 0]$ | $[2, 1]$ | $[3, 6]$ |