Defining polynomial over unramified subextension
| $x^{2} + d_{0} \pi$ |
Invariants
| Residue field characteristic: | $7$ |
| Degree: | $10$ |
| Base field: | $\Q_{7}(\sqrt{7\cdot 3})$ |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $5$ |
| Absolute Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $1$ (complete) |
| Ambiguity: | $10$ |
| Mass: | $1$ |
| Absolute Mass: | $1/10$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 7 }$ within this relative family, not the relative extension.
| Galois group: | $C_5\times D_4$ |
| Hidden Artin slopes: | $[\ ]^{2}$ |
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[2]$ |
| Jump Set: | undefined |
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 |
|---|---|---|---|---|---|---|---|---|
| 7.5.4.15a1.2 | $( x^{5} + x + 4 )^{4} + 7$ | $C_5\times D_4$ (as 20T12) | $40$ | $10$ | $[\ ]^{2}$ | $[0]$ | $[2]$ | undefined |