Defining polynomial
| $x^{5} + \pi$ |
Invariants
| Residue field characteristic: | $2$ |
| Degree: | $5$ |
| Base field: | $\Q_{2}(\sqrt{2})$ |
| Ramification index $e$: | $5$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $4$ |
| Absolute Artin slopes: | $[3]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $1$ (complete) |
| Ambiguity: | $1$ |
| Mass: | $1$ |
| Absolute Mass: | $1/2$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $F_{5}\times C_2$ |
| Hidden Artin slopes: | $[\ ]^{4}$ |
| Indices of inseparability: | $[10,0]$ |
| Associated inertia: | $[4,1]$ |
| Jump Set: | $[5,15]$ |
Fields
Showing all 1
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 |
|---|---|---|---|---|---|---|---|---|
| 2.1.10.19a1.9 | $x^{10} + 4 x^{5} + 2$ | $F_{5}\times C_2$ (as 10T5) | $40$ | $2$ | $[\ ]^{4}$ | $[10, 0]$ | $[4, 1]$ | $[5, 15]$ |