Defining polynomial over unramified subextension
| $x^{3} + 5d_{0}$ |
Invariants
| Residue field characteristic: | $5$ |
| Degree: | $12$ |
| Base field: | $\Q_{5}$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $4$ |
| Discriminant exponent $c$: | $8$ |
| Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $2$ (complete) |
| Ambiguity: | $12$ |
| Mass: | $1$ |
| Absolute Mass: | $1/4$ |
Varying
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[1]$ |
| Jump Set: | undefined |
Galois groups and Hidden Artin slopes
Select desired size of Galois group.
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 |
|---|---|---|---|---|---|---|---|---|
| 5.4.3.8a1.1 | $( x^{4} + 4 x^{2} + 4 x + 2 )^{3} + 5 x$ | $C_3\times (C_3 : C_4)$ (as 12T19) | $36$ | $6$ | $[\ ]^{3}$ | $[0]$ | $[1]$ | undefined |
| 5.4.3.8a1.2 | $( x^{4} + 4 x^{2} + 4 x + 2 )^{3} + 5$ | $C_3 : C_4$ (as 12T5) | $12$ | $12$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |