Defining polynomial
| $x^{5} + a_{1} \pi x + \pi$ |
Invariants
| Residue field characteristic: | $5$ |
| Degree: | $5$ |
| Base field: | 5.1.4.3a1.2 |
| Ramification index $e$: | $5$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $5$ |
| Absolute Artin slopes: | $[\frac{17}{16}]$ |
| Swan slopes: | $[\frac{1}{4}]$ |
| Means: | $\langle\frac{1}{5}\rangle$ |
| Rams: | $(\frac{1}{4})$ |
| Field count: | $1$ (complete) |
| Ambiguity: | $1$ |
| Mass: | $4$ |
| Absolute Mass: | $1$ |
Diagrams
Varying
These invariants are all associated to absolute extensions of $\Q_{ 5 }$ within this relative family, not the relative extension.
| Indices of inseparability: | $[1,0]$ |
| Associated inertia: | $[1,1]$ |
| Jump Set: | undefined |
Fields
Showing all 1
Download displayed columns for results| Label | Polynomial $/ \Q_p$ | Galois group $/ \Q_p$ | Galois degree $/ \Q_p$ | $\#\Aut(K/\Q_p)$ | Hidden Artin slopes $/ \Q_p$ | Ind. of Insep. $/ \Q_p$ | Assoc. Inertia $/ \Q_p$ | Jump Set |
|---|---|---|---|---|---|---|---|---|
| 5.1.20.20a1.2 | $x^{20} + 5 x + 10$ | not computed | $ $not computed$ $ | $1$ | not computed | $[1, 0]$ | $[1, 1]$ | undefined |