Defining polynomial
| $x^{9} + 3a_{2} x^{2} + 3$ |
Invariants
| Residue field characteristic: | $3$ |
| Degree: | $9$ |
| Base field: | $\Q_{3}$ |
| Ramification index $e$: | $9$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $10$ |
| Artin slopes: | $[\frac{5}{4},\frac{5}{4}]$ |
| Swan slopes: | $[\frac{1}{4},\frac{1}{4}]$ |
| Means: | $\langle\frac{1}{6},\frac{2}{9}\rangle$ |
| Rams: | $(\frac{1}{4},\frac{1}{4})$ |
| Field count: | $2$ (complete) |
| Ambiguity: | $1$ |
| Mass: | $2$ |
| Absolute Mass: | $2$ |
Diagrams
Varying
| Indices of inseparability: | $[2,2,0]$ |
| Associated inertia: | $[1]$ (show 1), $[2]$ (show 1) |
| Jump Set: | undefined |
Galois groups and Hidden Artin slopes
Fields
Showing all 2
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 |
|---|---|---|---|---|---|---|---|---|
| 3.1.9.10a1.1 | $x^{9} + 3 x^{2} + 3$ | $C_3^2:Q_8$ (as 9T14) | $72$ | $1$ | $[\ ]^{2}_{4}$ | $[2, 2, 0]$ | $[2]$ | undefined |
| 3.1.9.10a2.1 | $x^{9} + 6 x^{2} + 3$ | $S_3^2:C_2$ (as 9T16) | $72$ | $1$ | $[\ ]^{2}_{4}$ | $[2, 2, 0]$ | $[1]$ | undefined |