Defining polynomial over unramified subextension
| $x^{4} + 5d_{0}$ |
Invariants
| Residue field characteristic: | $5$ |
| Degree: | $20$ |
| Base field: | $\Q_{5}$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $15$ |
| Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $4$ (complete) |
| Ambiguity: | $20$ |
| Mass: | $1$ |
| Absolute Mass: | $1/5$ |
Varying
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[1]$ |
| Jump Set: | undefined (show 3), $[1]$ (show 1) |
Fields
Showing all 4
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 |
|---|---|---|---|---|---|---|---|---|
| 5.5.4.15a1.1 | $( x^{5} + 4 x + 3 )^{4} + 5 x^{3}$ | $C_{20}$ (as 20T1) | $20$ | $20$ | not computed | $[0]$ | $[1]$ | undefined |
| 5.5.4.15a1.2 | $( x^{5} + 4 x + 3 )^{4} + 5 x^{2}$ | $C_{20}$ (as 20T1) | $20$ | $20$ | not computed | $[0]$ | $[1]$ | undefined |
| 5.5.4.15a1.3 | $( x^{5} + 4 x + 3 )^{4} + 5 x$ | $C_{20}$ (as 20T1) | $20$ | $20$ | not computed | $[0]$ | $[1]$ | undefined |
| 5.5.4.15a1.4 | $( x^{5} + 4 x + 3 )^{4} + 5$ | $C_{20}$ (as 20T1) | $20$ | $20$ | not computed | $[0]$ | $[1]$ | $[1]$ |