Defining polynomial
| $x^{4} + d_{0} \pi$ |
Invariants
| Residue field characteristic: | $61$ |
| Degree: | $4$ |
| Base field: | $\Q_{61}(\sqrt[5]{61})$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $3$ |
| Absolute Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $4$ (complete) |
| Ambiguity: | $4$ |
| Mass: | $1$ |
| Absolute Mass: | $1/5$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 61 }$ within this relative family, not the relative extension.
| Galois group: | $C_{20}$ |
| Hidden Artin slopes: | $[\ ]$ |
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[1]$ |
| Jump Set: | undefined |
Fields
Showing all 4
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 |
|---|---|---|---|---|---|---|---|---|
| 61.1.20.19a1.1 | $x^{20} + 61$ | $C_{20}$ (as 20T1) | $20$ | $20$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 61.1.20.19a1.8 | $x^{20} + 671$ | $C_{20}$ (as 20T1) | $20$ | $20$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 61.1.20.19a1.15 | $x^{20} + 1952$ | $C_{20}$ (as 20T1) | $20$ | $20$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 61.1.20.19a1.19 | $x^{20} + 2928$ | $C_{20}$ (as 20T1) | $20$ | $20$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |