Defining polynomial over unramified subextension
| $x^{2} + 2 a_{1} x + 4 c_{2} + 2$ |
Invariants
| Residue field characteristic: | $2$ |
| Degree: | $22$ |
| Base field: | $\Q_{2}$ |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $11$ |
| Discriminant exponent $c$: | $22$ |
| Artin slopes: | $[2]$ |
| Swan slopes: | $[1]$ |
| Means: | $\langle\frac{1}{2}\rangle$ |
| Rams: | $(1)$ |
| Field count: | $374$ (complete) |
| Ambiguity: | $22$ |
| Mass: | $2047$ |
| Absolute Mass: | $2047/11$ |
Diagrams
Varying
| Indices of inseparability: | $[1,0]$ |
| Associated inertia: | $[1]$ |
| Jump Set: | $[1,2]$ (show 1), $[1,3]$ (show 372), $[1,4]$ (show 1) |
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 |
|---|---|---|---|---|---|---|---|---|
| 2.11.2.22a1.2 | $( x^{11} + x^{2} + 1 )^{2} + 2 ( x^{11} + x^{2} + 1 ) + 6$ | $C_{22}$ (as 22T1) | $22$ | $22$ | not computed | $[1, 0]$ | $[1]$ | $[1, 4]$ |