Defining polynomial over unramified subextension
| $x^{2} + d_{0} \pi$ |
Invariants
| Residue field characteristic: | $3$ |
| Degree: | $6$ |
| Base field: | 3.1.3.4a2.3 |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $3$ |
| Absolute Artin slopes: | $[2]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $2$ (complete) |
| Ambiguity: | $6$ |
| Mass: | $1$ |
| Absolute Mass: | $1/9$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.
| Indices of inseparability: | $[4,0]$ |
| Associated inertia: | $[1,1]$ |
| Jump Set: | undefined (show 1), $[1,3]$ (show 1) |
Fields
Showing all 2
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 |
|---|---|---|---|---|---|---|---|---|
| 3.3.6.27a1.43 | $( x^{3} + 2 x + 1 )^{6} + 6 ( x^{3} + 2 x + 1 )^{4} + 3$ | not computed | $ $not computed$ $ | $18$ | not computed | $[4, 0]$ | $[1, 1]$ | $[1, 3]$ |
| 3.3.6.27a12.1 | $( x^{3} + 2 x + 1 )^{6} + \left(6 x + 6\right) ( x^{3} + 2 x + 1 )^{4} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ | not computed | $ $not computed$ $ | $18$ | not computed | $[4, 0]$ | $[1, 1]$ | undefined |