Defining polynomial over unramified subextension
| $x^{9} + 2d_{0}$ |
Invariants
| Residue field characteristic: | $2$ |
| Degree: | $18$ |
| Base field: | $\Q_{2}$ |
| Ramification index $e$: | $9$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $16$ |
| Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $2$ (complete) |
| Ambiguity: | $6$ |
| Mass: | $1$ |
| Absolute Mass: | $1/2$ |
Varying
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[3]$ |
| Jump Set: | $[9]$ |
Galois groups and Hidden Artin slopes
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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 |
|---|---|---|---|---|---|---|---|---|
| 2.2.9.16a1.1 | $( x^{2} + x + 1 )^{9} + 2 x$ | $C_9:C_{18}$ (as 18T80) | $162$ | $3$ | $[\ ]^{9}$ | $[0]$ | $[3]$ | $[9]$ |
| 2.2.9.16a1.2 | $( x^{2} + x + 1 )^{9} + 2$ | $C_9:C_6$ (as 18T18) | $54$ | $6$ | $[\ ]^{3}$ | $[0]$ | $[3]$ | $[9]$ |