Defining polynomial over unramified subextension
| $x^{4} + 5d_{0}$ |
Invariants
| Residue field characteristic: | $5$ |
| Degree: | $8$ |
| Base field: | $\Q_{5}(\sqrt{2})$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $6$ |
| Absolute Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $4$ (complete) |
| Ambiguity: | $8$ |
| Mass: | $1$ |
| Absolute Mass: | $1/4$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 5 }$ within this relative family, not the relative extension.
| Galois group: | $C_{16}$ (show 2), $C_4^2$ (show 1), $C_8\times C_2$ (show 1) |
| Hidden Artin slopes: | $[\ ]$ |
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[1]$ |
| Jump Set: | undefined (show 3), $[1]$ (show 1) |
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 |
|---|---|---|---|---|---|---|---|---|
| 5.4.4.12a1.1 | $( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5 x^{3}$ | $C_{16}$ (as 16T1) | $16$ | $16$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 5.4.4.12a1.2 | $( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5 x^{2}$ | $C_8\times C_2$ (as 16T5) | $16$ | $16$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 5.4.4.12a1.3 | $( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5 x$ | $C_{16}$ (as 16T1) | $16$ | $16$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 5.4.4.12a1.4 | $( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ | $C_4^2$ (as 16T4) | $16$ | $16$ | $[\ ]$ | $[0]$ | $[1]$ | $[1]$ |