Defining polynomial
| $x^{6} + 3 a_{1} x + 3 d_{0}$ |
Invariants
| Residue field characteristic: | $3$ |
| Degree: | $6$ |
| Base field: | $\Q_{3}$ |
| Ramification index $e$: | $6$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $6$ |
| Artin slopes: | $[\frac{5}{4}]$ |
| Swan slopes: | $[\frac{1}{4}]$ |
| Means: | $\langle\frac{1}{6}\rangle$ |
| Rams: | $(\frac{1}{2})$ |
| Field count: | $2$ (complete) |
| Ambiguity: | $2$ |
| Mass: | $2$ |
| Absolute Mass: | $2$ |
Diagrams
Varying
| Indices of inseparability: | $[1,0]$ |
| Associated inertia: | $[1,1]$ |
| Jump Set: | undefined (show 1), $[1,4]$ (show 1) |
Galois groups and Hidden Artin slopes
Fields
Showing all 1
Download displayed columns for results| Label | Polynomial | Galois group | Galois degree | $\#\Aut(K/\Q_p)$ | Hidden Artin slopes | Ind. of Insep. | Assoc. Inertia | Jump Set |
|---|---|---|---|---|---|---|---|---|
| 3.1.6.6a1.1 | $x^{6} + 3 x + 3$ | $C_3^2:D_4$ (as 6T13) | $72$ | $1$ | $[\frac{5}{4}]^{2}_{2}$ | $[1, 0]$ | $[1, 1]$ | $[1, 4]$ |