Defining polynomial over unramified subextension
| $x^{3} + d_{0} \pi$ |
Invariants
| Residue field characteristic: | $181$ |
| Degree: | $6$ |
| Base field: | $\Q_{181}(\sqrt{181})$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $4$ |
| Absolute Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $3$ (complete) |
| Ambiguity: | $6$ |
| Mass: | $1$ |
| Absolute Mass: | $1/4$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 181 }$ within this relative family, not the relative extension.
| Galois group: | $C_6\times C_2$ |
| Hidden Artin slopes: | $[\ ]$ |
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[1]$ |
| Jump Set: | undefined |
Fields
Showing all 3
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 |
|---|---|---|---|---|---|---|---|---|
| 181.2.6.10a1.2 | $( x^{2} + 177 x + 2 )^{6} + 181$ | $C_6\times C_2$ (as 12T2) | $12$ | $12$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 181.2.6.10a1.4 | $( x^{2} + 177 x + 2 )^{6} + 8688 x + 27693$ | $C_6\times C_2$ (as 12T2) | $12$ | $12$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 181.2.6.10a1.6 | $( x^{2} + 177 x + 2 )^{6} + 724 x + 32399$ | $C_6\times C_2$ (as 12T2) | $12$ | $12$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |