Defining polynomial
| $x^{4} + d_{0} \pi$ |
Invariants
| Residue field characteristic: | $3$ |
| Degree: | $4$ |
| Base field: | 3.1.3.4a2.3 |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $3$ |
| Absolute Artin slopes: | $[2]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $2$ (complete) |
| Ambiguity: | $2$ |
| Mass: | $1$ |
| Absolute Mass: | $1/3$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.
| Galois group: | $D_4 \times C_3$ |
| Hidden Artin slopes: | $[\ ]^{2}$ |
| Indices of inseparability: | $[8,0]$ |
| Associated inertia: | $[2,1]$ |
| Jump Set: | undefined (show 1), $[2,18]$ (show 1) |
Fields
Showing all 2
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 |
|---|---|---|---|---|---|---|---|---|
| 3.1.12.19a2.3 | $x^{12} + 6 x^{8} + 21$ | $D_4 \times C_3$ (as 12T14) | $24$ | $6$ | $[\ ]^{2}$ | $[8, 0]$ | $[2, 1]$ | $[2, 18]$ |
| 3.1.12.19a2.19 | $x^{12} + 3 x^{8} + 6$ | $D_4 \times C_3$ (as 12T14) | $24$ | $6$ | $[\ ]^{2}$ | $[8, 0]$ | $[2, 1]$ | undefined |