Defining polynomial
| $x^{20} + 7d_{0}$ |
Invariants
| Residue field characteristic: | $7$ |
| Degree: | $20$ |
| Base field: | $\Q_{7}$ |
| Ramification index $e$: | $20$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $19$ |
| Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $2$ (complete) |
| Ambiguity: | $2$ |
| Mass: | $1$ |
| Absolute Mass: | $1$ |
Varying
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[4]$ |
| Jump Set: | undefined |
Galois groups and Hidden Artin slopes
Fields
Showing all 2
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 |
|---|---|---|---|---|---|---|---|---|
| 7.1.20.19a1.1 | $x^{20} + 7$ | $C_{20}:C_4$ (as 20T18) | $80$ | $2$ | $[\ ]^{4}$ | $[0]$ | $[4]$ | undefined |
| 7.1.20.19a1.2 | $x^{20} + 21$ | $C_{20}:C_4$ (as 20T18) | $80$ | $2$ | $[\ ]^{4}$ | $[0]$ | $[4]$ | undefined |