Defining polynomial
| $x^{7} + 197d_{0}$ |
Invariants
| Residue field characteristic: | $197$ |
| Degree: | $7$ |
| Base field: | $\Q_{197}$ |
| Ramification index $e$: | $7$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $6$ |
| Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $7$ (complete) |
| Ambiguity: | $7$ |
| Mass: | $1$ |
| Absolute Mass: | $1$ |
Varying
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[1]$ |
| Jump Set: | undefined |
Galois groups and Hidden Artin slopes
Fields
Showing all 7
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 |
|---|---|---|---|---|---|---|---|---|
| 197.1.7.6a1.1 | $x^{7} + 197$ | $C_7$ (as 7T1) | $7$ | $7$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 197.1.7.6a1.2 | $x^{7} + 394$ | $C_7$ (as 7T1) | $7$ | $7$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 197.1.7.6a1.3 | $x^{7} + 788$ | $C_7$ (as 7T1) | $7$ | $7$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 197.1.7.6a1.4 | $x^{7} + 1576$ | $C_7$ (as 7T1) | $7$ | $7$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 197.1.7.6a1.5 | $x^{7} + 3152$ | $C_7$ (as 7T1) | $7$ | $7$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 197.1.7.6a1.6 | $x^{7} + 6304$ | $C_7$ (as 7T1) | $7$ | $7$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 197.1.7.6a1.7 | $x^{7} + 12608$ | $C_7$ (as 7T1) | $7$ | $7$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |