Defining polynomial over unramified subextension
| $x^{2} + a_{1} \pi x + c_{2} \pi^{2} + \pi$ |
Invariants
| Residue field characteristic: | $2$ |
| Degree: | $4$ |
| Base field: | $\Q_{2}(\sqrt{-1})$ |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $4$ |
| Absolute Artin slopes: | $[2,2]$ |
| Swan slopes: | $[1]$ |
| Means: | $\langle\frac{1}{2}\rangle$ |
| Rams: | $(1)$ |
| Field count: | $3$ (complete) |
| Ambiguity: | $4$ |
| Mass: | $3$ |
| Absolute Mass: | $3/4$ |
Diagrams
Varying
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $D_4$ (show 2), $C_2^3: C_4$ (show 1) |
| Hidden Artin slopes: | $[\ ]$ (show 2), $[2]^{2}$ (show 1) |
| Indices of inseparability: | $[3,2,0]$ (show 1), $[3,3,0]$ (show 2) |
| Associated inertia: | $[1]$ (show 2), $[2]$ (show 1) |
| Jump Set: | $[1,3,6]$ |
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 |
|---|---|---|---|---|---|---|---|---|
| 2.2.4.12a6.1 | $( x^{2} + x + 1 )^{4} + \left(2 x + 2\right) ( x^{2} + x + 1 )^{3} + 2 x ( x^{2} + x + 1 )^{2} + 2$ | $C_2^3: C_4$ (as 8T21) | $32$ | $2$ | $[2]^{2}$ | $[3, 2, 0]$ | $[2]$ | $[1, 3, 6]$ |