Defining polynomial
| $x^{5} + 19$ |
Invariants
| Residue field characteristic: | $19$ |
| Degree: | $5$ |
| Base field: | $\Q_{19}$ |
| Ramification index $e$: | $5$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $4$ |
| Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $1$ (complete) |
| Ambiguity: | $1$ |
| Mass: | $1$ |
| Absolute Mass: | $1$ |
Varying
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[2]$ |
| Jump Set: | undefined |
Galois groups and Hidden Artin slopes
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 |
|---|---|---|---|---|---|---|---|---|
| 19.1.5.4a1.1 | $x^{5} + 19$ | $D_{5}$ (as 5T2) | $10$ | $1$ | $[\ ]^{2}$ | $[0]$ | $[2]$ | undefined |