Defining polynomial
| $x^{16} + 197d_{0}$ |
Invariants
| Residue field characteristic: | $197$ |
| Degree: | $16$ |
| Base field: | $\Q_{197}$ |
| Ramification index $e$: | $16$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $15$ |
| Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $4$ (complete) |
| Ambiguity: | $4$ |
| 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 4
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 |
|---|---|---|---|---|---|---|---|---|
| 197.1.16.15a1.1 | $x^{16} + 197$ | $C_{16}:C_4$ (as 16T125) | $64$ | $4$ | $[\ ]^{4}$ | $[0]$ | $[4]$ | undefined |
| 197.1.16.15a1.2 | $x^{16} + 394$ | $C_{16}:C_4$ (as 16T125) | $64$ | $4$ | $[\ ]^{4}$ | $[0]$ | $[4]$ | undefined |
| 197.1.16.15a1.3 | $x^{16} + 788$ | $C_{16}:C_4$ (as 16T125) | $64$ | $4$ | $[\ ]^{4}$ | $[0]$ | $[4]$ | undefined |
| 197.1.16.15a1.4 | $x^{16} + 1576$ | $C_{16}:C_4$ (as 16T125) | $64$ | $4$ | $[\ ]^{4}$ | $[0]$ | $[4]$ | undefined |