Defining polynomial
| $x^{2} + d_{0} \pi$ |
Invariants
| Residue field characteristic: | $3$ |
| Degree: | $2$ |
| Base field: | 3.1.3.4a2.1 |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $1$ |
| 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: | $C_6$ |
| Hidden Artin slopes: | $[\ ]$ |
| Indices of inseparability: | $[4,0]$ |
| Associated inertia: | $[1,1]$ |
| Jump Set: | undefined (show 1), $[1,3]$ (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 |
|---|---|---|---|---|---|---|---|---|
| 3.1.6.9a1.1 | $x^{6} + 6 x^{4} + 3$ | $C_6$ (as 6T1) | $6$ | $6$ | $[\ ]$ | $[4, 0]$ | $[1, 1]$ | $[1, 3]$ |
| 3.1.6.9a1.9 | $x^{6} + 3 x^{4} + 24$ | $C_6$ (as 6T1) | $6$ | $6$ | $[\ ]$ | $[4, 0]$ | $[1, 1]$ | undefined |