Defining polynomial
| $x^{5} + \pi$ |
Invariants
| Residue field characteristic: | $109$ |
| Degree: | $5$ |
| Base field: | 109.1.4.3a1.1 |
| Ramification index $e$: | $5$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $4$ |
| Absolute Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $1$ (complete) |
| Ambiguity: | $1$ |
| Mass: | $1$ |
| Absolute Mass: | $1/4$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 109 }$ within this relative family, not the relative extension.
| Galois group: | $C_4\times D_5$ |
| Hidden Artin slopes: | $[\ ]^{2}$ |
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[2]$ |
| 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 |
|---|---|---|---|---|---|---|---|---|
| 109.1.20.19a1.1 | $x^{20} + 109$ | $C_4\times D_5$ (as 20T6) | $40$ | $4$ | $[\ ]^{2}$ | $[0]$ | $[2]$ | undefined |