Defining polynomial
| $x^{6} + 191d_{0}$ |
Invariants
| Residue field characteristic: | $191$ |
| Degree: | $6$ |
| Base field: | $\Q_{191}(\sqrt{7})$ |
| Ramification index $e$: | $6$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $5$ |
| Absolute Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $4$ (complete) |
| Ambiguity: | $6$ |
| Mass: | $1$ |
| Absolute Mass: | $1/2$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 191 }$ within this relative family, not the relative extension.
| Galois group: | $D_6$ (show 1), $C_3 : C_4$ (show 1), $C_6\times S_3$ (show 1), $C_3\times (C_3 : C_4)$ (show 1) |
| Hidden Artin slopes: | $[\ ]$ (show 2), $[\ ]^{3}$ (show 2) |
| 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 |
|---|---|---|---|---|---|---|---|---|
| 191.2.6.10a1.1 | $( x^{2} + 190 x + 19 )^{6} + 191 x$ | $C_3\times (C_3 : C_4)$ (as 12T19) | $36$ | $6$ | $[\ ]^{3}$ | $[0]$ | $[1]$ | undefined |
| 191.2.6.10a1.2 | $( x^{2} + 190 x + 19 )^{6} + 191$ | $D_6$ (as 12T3) | $12$ | $12$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 191.2.6.10a1.3 | $( x^{2} + 190 x + 19 )^{6} + 191 x + 32852$ | $C_6\times S_3$ (as 12T18) | $36$ | $6$ | $[\ ]^{3}$ | $[0]$ | $[1]$ | undefined |
| 191.2.6.10a1.4 | $( x^{2} + 190 x + 19 )^{6} + 33043 x + 32852$ | $C_3 : C_4$ (as 12T5) | $12$ | $12$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |