Defining polynomial
| $x^{3} + b_{8} \pi^{3} x^{2} + b_{7} \pi^{3} x + c_{9} \pi^{4} + \pi$ |
Invariants
| Residue field characteristic: | $3$ |
| Degree: | $3$ |
| Base field: | $\Q_{3}(\sqrt{3})$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $8$ |
| Absolute Artin slopes: | $[\frac{5}{2}]$ |
| Swan slopes: | $[3]$ |
| Means: | $\langle2\rangle$ |
| Rams: | $(3)$ |
| Field count: | $6$ (complete) |
| Ambiguity: | $3$ |
| Mass: | $9$ |
| Absolute Mass: | $9/2$ |
Diagrams
Varying
These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.
| Galois group: | $D_{6}$ (show 3), $S_3^2$ (show 3) |
| Hidden Artin slopes: | $[\ ]^{2}$ (show 3), $[2]^{2}$ (show 3) |
| Indices of inseparability: | $[6,0]$ |
| Associated inertia: | $[1,2]$ |
| Jump Set: | undefined |
Fields
Showing all 3
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 |
|---|---|---|---|---|---|---|---|---|
| 3.1.6.11a2.1 | $x^{6} + 6$ | $D_{6}$ (as 6T3) | $12$ | $2$ | $[\ ]^{2}$ | $[6, 0]$ | $[1, 2]$ | undefined |
| 3.1.6.11a2.2 | $x^{6} + 9 x^{2} + 6$ | $D_{6}$ (as 6T3) | $12$ | $2$ | $[\ ]^{2}$ | $[6, 0]$ | $[1, 2]$ | undefined |
| 3.1.6.11a2.3 | $x^{6} + 18 x^{2} + 6$ | $D_{6}$ (as 6T3) | $12$ | $2$ | $[\ ]^{2}$ | $[6, 0]$ | $[1, 2]$ | undefined |