Defining polynomial
| $x^{11} + 11a_{6} x^{6} + 11$ |
Invariants
| Residue field characteristic: | $11$ |
| Degree: | $11$ |
| Base field: | $\Q_{11}$ |
| Ramification index $e$: | $11$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $16$ |
| Artin slopes: | $[\frac{8}{5}]$ |
| Swan slopes: | $[\frac{3}{5}]$ |
| Means: | $\langle\frac{6}{11}\rangle$ |
| Rams: | $(\frac{3}{5})$ |
| Field count: | $10$ (complete) |
| Ambiguity: | $1$ |
| Mass: | $10$ |
| Absolute Mass: | $10$ |
Diagrams
Varying
| Indices of inseparability: | $[6,0]$ |
| Associated inertia: | $[1]$ (show 5), $[2]$ (show 5) |
| Jump Set: | undefined |
Galois groups and Hidden Artin slopes
Select desired size of Galois group.
Fields
Showing all 5
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 |
|---|---|---|---|---|---|---|---|---|
| 11.1.11.16a1.1 | $x^{11} + 11 x^{6} + 11$ | $F_{11}$ (as 11T4) | $110$ | $1$ | $[\ ]^{2}_{5}$ | $[6, 0]$ | $[2]$ | undefined |
| 11.1.11.16a1.2 | $x^{11} + 33 x^{6} + 11$ | $F_{11}$ (as 11T4) | $110$ | $1$ | $[\ ]^{2}_{5}$ | $[6, 0]$ | $[2]$ | undefined |
| 11.1.11.16a1.3 | $x^{11} + 44 x^{6} + 11$ | $F_{11}$ (as 11T4) | $110$ | $1$ | $[\ ]^{2}_{5}$ | $[6, 0]$ | $[2]$ | undefined |
| 11.1.11.16a1.4 | $x^{11} + 55 x^{6} + 11$ | $F_{11}$ (as 11T4) | $110$ | $1$ | $[\ ]^{2}_{5}$ | $[6, 0]$ | $[2]$ | undefined |
| 11.1.11.16a1.5 | $x^{11} + 99 x^{6} + 11$ | $F_{11}$ (as 11T4) | $110$ | $1$ | $[\ ]^{2}_{5}$ | $[6, 0]$ | $[2]$ | undefined |