Defining polynomial over unramified subextension
| $x^{4} + 3d_{0}$ |
Invariants
| Residue field characteristic: | $3$ |
| Degree: | $20$ |
| Base field: | $\Q_{3}$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $15$ |
| Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $2$ (complete) |
| Ambiguity: | $10$ |
| Mass: | $1$ |
| Absolute Mass: | $1/5$ |
Varying
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[2]$ |
| Jump Set: | undefined (show 1), $[2]$ (show 1) |
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 |
|---|---|---|---|---|---|---|---|---|
| 3.5.4.15a1.1 | $( x^{5} + 2 x + 1 )^{4} + 3 x$ | $C_5\times D_4$ (as 20T12) | $40$ | $10$ | $[\ ]^{2}$ | $[0]$ | $[2]$ | undefined |
| 3.5.4.15a1.2 | $( x^{5} + 2 x + 1 )^{4} + 3$ | $C_5\times D_4$ (as 20T12) | $40$ | $10$ | $[\ ]^{2}$ | $[0]$ | $[2]$ | $[2]$ |