Defining polynomial over unramified subextension
| $x^{6} + 3 b_{5} x^{5} + 3 a_{4} x^{4} + 3 d_{0} + 9 c_{6}$ |
Invariants
| Residue field characteristic: | $3$ |
| Degree: | $18$ |
| Base field: | $\Q_{3}$ |
| Ramification index $e$: | $6$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $27$ |
| Artin slopes: | $[2]$ |
| Swan slopes: | $[1]$ |
| Means: | $\langle\frac{2}{3}\rangle$ |
| Rams: | $(2)$ |
| Field count: | $496$ (complete) |
| Ambiguity: | $18$ |
| Mass: | $702$ |
| Absolute Mass: | $234$ |
Diagrams
Varying
| Indices of inseparability: | $[4,0]$ |
| Associated inertia: | $[1,1]$ (show 372), $[1,2]$ (show 124) |
| Jump Set: | undefined (show 248), $[1,3]$ (show 1), $[1,7]$ (show 230), $[1,8]$ (show 15), $[1,9]$ (show 2) |
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 |
|---|---|---|---|---|---|---|---|---|
| 3.3.6.27a1.44 | $( x^{3} + 2 x + 1 )^{6} + 6 ( x^{3} + 2 x + 1 )^{4} + 9 x^{2} + 3$ | not computed | $ $not computed$ $ | $18$ | not computed | $[4, 0]$ | $[1, 1]$ | $[1, 9]$ |
| 3.3.6.27a1.45 | $( x^{3} + 2 x + 1 )^{6} + 6 ( x^{3} + 2 x + 1 )^{4} + 18 x^{2} + 3$ | $C_{18}$ (as 18T1) | $18$ | $18$ | not computed | $[4, 0]$ | $[1, 1]$ | $[1, 9]$ |