Defining polynomial
| $x^{3} + 3 a_{2} x^{2} + 9 c_{3} + 3$ |
Invariants
| Residue field characteristic: | $3$ |
| Degree: | $3$ |
| Base field: | $\Q_{3}$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $4$ |
| Artin slopes: | $[2]$ |
| Swan slopes: | $[1]$ |
| Means: | $\langle\frac{2}{3}\rangle$ |
| Rams: | $(1)$ |
| Field count: | $4$ (complete) |
| Ambiguity: | $3$ |
| Mass: | $2$ |
| Absolute Mass: | $2$ |
Diagrams
Varying
| Indices of inseparability: | $[2,0]$ |
| Associated inertia: | $[1]$ (show 3), $[2]$ (show 1) |
| Jump Set: | undefined |
Galois groups and Hidden Artin slopes
Select desired size of Galois group.
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 |
|---|---|---|---|---|---|---|---|---|
| 3.1.3.4a1.1 | $x^{3} + 3 x^{2} + 3$ | $S_3$ (as 3T2) | $6$ | $1$ | $[\ ]^{2}$ | $[2, 0]$ | $[2]$ | undefined |