Defining polynomial
| $x^{3} + a_{2} \pi x^{2} + c_{3} \pi^{2} + \pi$ |
Invariants
| Residue field characteristic: | $3$ |
| Degree: | $3$ |
| Base field: | $\Q_{3}(\sqrt{3})$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $4$ |
| Absolute Artin slopes: | $[\frac{3}{2}]$ |
| Swan slopes: | $[1]$ |
| Means: | $\langle\frac{2}{3}\rangle$ |
| Rams: | $(1)$ |
| Field count: | $3$ (complete) |
| Ambiguity: | $3$ |
| 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: | $S_3$ (show 1), $D_{6}$ (show 1), $S_3\times C_3$ (show 1) |
| Hidden Artin slopes: | $[\ ]$ (show 1), $[\ ]^{2}$ (show 1), $[\ ]^{3}$ (show 1) |
| Indices of inseparability: | $[2,0]$ |
| Associated inertia: | $[1,1]$ (show 2), $[1,2]$ (show 1) |
| 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 |
|---|---|---|---|---|---|---|---|---|
| 3.1.6.7a1.3 | $x^{6} + 6 x^{2} + 6$ | $S_3$ (as 6T2) | $6$ | $6$ | $[\ ]$ | $[2, 0]$ | $[1, 1]$ | undefined |
| 3.1.6.7a1.4 | $x^{6} + 3 x^{3} + 6 x^{2} + 6$ | $S_3\times C_3$ (as 6T5) | $18$ | $3$ | $[\ ]^{3}$ | $[2, 0]$ | $[1, 1]$ | undefined |
| 3.1.6.7a2.2 | $x^{6} + 3 x^{2} + 6$ | $D_{6}$ (as 6T3) | $12$ | $2$ | $[\ ]^{2}$ | $[2, 0]$ | $[1, 2]$ | undefined |