Invariants
| Residue field characteristic: | $3$ |
| Degree: | $2$ |
| Base field: | 3.1.6.11a1.7 |
| Ramification index $e$: | $1$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $0$ |
| Absolute Artin slopes: | $[\frac{5}{2}]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $1$ (complete) |
| Ambiguity: | $2$ |
| Mass: | $1$ |
| Absolute Mass: | $1/6$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.
| Galois group: | $C_6\times S_3$ |
| Hidden Artin slopes: | $[2]$ |
| Indices of inseparability: | $[6,0]$ |
| Associated inertia: | $[1,1]$ |
| Jump Set: | $[1,7]$ |
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 |
|---|---|---|---|---|---|---|---|---|
| 3.2.6.22a1.40 | $( x^{2} + 2 x + 2 )^{6} + 9 ( x^{2} + 2 x + 2 ) + 3$ | $C_6\times S_3$ (as 12T18) | $36$ | $6$ | $[2]$ | $[6, 0]$ | $[1, 1]$ | $[1, 7]$ |