Defining polynomial
| $x^{3} + 3 a_{2} x^{2} + 9 c_{3} + 3$ |
Invariants
| Residue field characteristic: | $3$ |
| Degree: | $3$ |
| Base field: | $\Q_{3}(\sqrt{2})$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $4$ |
| Absolute Artin slopes: | $[2]$ |
| Swan slopes: | $[1]$ |
| Means: | $\langle\frac{2}{3}\rangle$ |
| Rams: | $(1)$ |
| Field count: | $10$ (complete) |
| Ambiguity: | $3$ |
| Mass: | $8$ |
| Absolute Mass: | $4$ |
Diagrams
Varying
These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.
| Galois group: | $C_6$ (show 3), $S_3$ (show 1), $S_3\times C_3$ (show 4), $C_3^2:C_4$ (show 2) |
| Hidden Artin slopes: | $[2]^{2}$ (show 2), $[\ ]$ (show 4), $[\ ]^{3}$ (show 1), $[2]$ (show 3) |
| Indices of inseparability: | $[2,0]$ |
| Associated inertia: | $[1]$ (show 8), $[2]$ (show 2) |
| Jump Set: | undefined |
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 |
|---|---|---|---|---|---|---|---|---|
| 3.2.3.8a1.2 | $( x^{2} + 2 x + 2 )^{3} + 3 ( x^{2} + 2 x + 2 )^{2} + 9 x + 3$ | $S_3\times C_3$ (as 6T5) | $18$ | $3$ | $[\ ]^{3}$ | $[2, 0]$ | $[1]$ | undefined |