Invariants
| Residue field characteristic: | $3$ |
| Degree: | $2$ |
| Base field: | $\Q_{3}(\zeta_{8}, \sqrt[4]{\zeta_{8} \cdot 3})$ |
| Ramification index $e$: | $1$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $0$ |
| Absolute Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $1$ (incomplete) |
| Ambiguity: | $2$ |
| Mass: | $1$ |
| Absolute Mass: | $1/8$ ($1/16$ currently in the LMFDB) |
Varying
The following invariants arise for fields within the LMFDB; since not all fields in this family are stored, it may be incomplete.
These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.
| Galois group: | $C_8: C_2$ (incomplete) |
| Hidden Artin slopes: | $[\ ]$ (incomplete) |
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[1]$ |
| Jump Set: | $[2]$ |
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 |
|---|---|---|---|---|---|---|---|---|
| 3.4.4.12a1.1 | $( x^{4} + 2 x^{3} + 2 )^{4} + 3 x^{2}$ | $C_8: C_2$ (as 16T6) | $16$ | $16$ | $[\ ]$ | $[0]$ | $[1]$ | $[2]$ |