Defining polynomial
| $x^{2} + d_{0} \pi$ |
Invariants
| Residue field characteristic: | $3$ |
| Degree: | $2$ |
| Base field: | 3.1.3.5a1.1 |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $1$ |
| Absolute Artin slopes: | $[\frac{5}{2}]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $2$ (incomplete) |
| Ambiguity: | $2$ |
| Mass: | $1$ |
| Absolute Mass: | $1$ ($2/3$ currently in the LMFDB) |
Varying
The following invariants arise for fields within the LMFDB; since not all fields in this family are stored, it may be incomplete.
These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.
| Galois group: | $S_3$ (show 1), $D_{6}$ (show 1) (incomplete) |
| Hidden Artin slopes: | $[\ ]^{2}$ (show 1), $[\ ]$ (show 1) (incomplete) |
| Indices of inseparability: | $[6,0]$ |
| Associated inertia: | $[1,1]$ (show 1), $[1,2]$ (show 1) |
| Jump Set: | undefined (show 1), $[1,7]$ (show 1) |
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 |
|---|---|---|---|---|---|---|---|---|
| 3.1.6.11a1.1 | $x^{6} + 3$ | $S_3$ (as 6T2) | $6$ | $6$ | $[\ ]$ | $[6, 0]$ | $[1, 1]$ | $[1, 7]$ |