Defining polynomial
| $x^{3} + \pi$ |
Invariants
| Residue field characteristic: | $2$ |
| Degree: | $3$ |
| Base field: | 2.3.2.9a1.6 |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $2$ |
| Absolute Artin slopes: | $[3]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $1$ (complete) |
| Ambiguity: | $1$ |
| Mass: | $1$ |
| Absolute Mass: | $1/6$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $S_3 \times C_6$ |
| Hidden Artin slopes: | $[\ ]^{2}$ |
| Indices of inseparability: | $[6,0]$ |
| Associated inertia: | $[2,1]$ |
| Jump Set: | $[3,9]$ |
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 |
|---|---|---|---|---|---|---|---|---|
| 2.3.6.33a1.26 | $( x^{3} + x + 1 )^{6} + 4 ( x^{3} + x + 1 )^{3} + 10$ | $S_3 \times C_6$ (as 18T6) | $36$ | $6$ | $[\ ]^{2}$ | $[6, 0]$ | $[2, 1]$ | $[3, 9]$ |