Defining polynomial
| $x^{20} + 5 a_{3} x^{3} + 5 d_{0}$ |
Invariants
| Residue field characteristic: | $5$ |
| Degree: | $20$ |
| Base field: | $\Q_{5}$ |
| Ramification index $e$: | $20$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $22$ |
| Artin slopes: | $[\frac{19}{16}]$ |
| Swan slopes: | $[\frac{3}{16}]$ |
| Means: | $\langle\frac{3}{20}\rangle$ |
| Rams: | $(\frac{3}{4})$ |
| Field count: | $4$ (complete) |
| Ambiguity: | $4$ |
| Mass: | $4$ |
| Absolute Mass: | $4$ |
Diagrams
Varying
| Indices of inseparability: | $[3,0]$ |
| Associated inertia: | $[1,1]$ |
| Jump Set: | undefined (show 3), $[1,8]$ (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 |
|---|---|---|---|---|---|---|---|---|
| 5.1.20.22a1.1 | $x^{20} + 5 x^{3} + 5$ | not computed | $ $not computed$ $ | $1$ | not computed | $[3, 0]$ | $[1, 1]$ | $[1, 8]$ |