Defining polynomial
| $x^{3} + a_{1} \pi x + \pi$ |
Invariants
| Residue field characteristic: | $3$ |
| Degree: | $3$ |
| Base field: | $\Q_{3}(\sqrt{3})$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $3$ |
| Absolute Artin slopes: | $[\frac{5}{4}]$ |
| Swan slopes: | $[\frac{1}{2}]$ |
| Means: | $\langle\frac{1}{3}\rangle$ |
| Rams: | $(\frac{1}{2})$ |
| Field count: | $1$ (complete) |
| Ambiguity: | $1$ |
| Mass: | $2$ |
| Absolute Mass: | $1$ |
Diagrams
Varying
These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.
| Galois group: | $C_3^2:D_4$ |
| Hidden Artin slopes: | $[\frac{5}{4}]^{2}_{2}$ |
| Indices of inseparability: | $[1,0]$ |
| Associated inertia: | $[1,1]$ |
| Jump Set: | undefined |
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 |
|---|---|---|---|---|---|---|---|---|
| 3.1.6.6a1.2 | $x^{6} + 3 x + 6$ | $C_3^2:D_4$ (as 6T13) | $72$ | $1$ | $[\frac{5}{4}]^{2}_{2}$ | $[1, 0]$ | $[1, 1]$ | undefined |