Defining polynomial
| $x^{2} + 19d_{0}$ |
Invariants
| Residue field characteristic: | $19$ |
| Degree: | $2$ |
| Base field: | 19.6.1.0a1.1 |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $1$ |
| Absolute Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $2$ (complete) |
| Ambiguity: | $2$ |
| Mass: | $1$ |
| Absolute Mass: | $1/6$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 19 }$ within this relative family, not the relative extension.
| Galois group: | $C_{12}$ (show 1), $C_6\times C_2$ (show 1) |
| Hidden Artin slopes: | $[\ ]$ |
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[1]$ |
| Jump Set: | undefined |
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 |
|---|---|---|---|---|---|---|---|---|
| 19.6.2.6a1.1 | $( x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2 )^{2} + 19 x$ | $C_{12}$ (as 12T1) | $12$ | $12$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 19.6.2.6a1.2 | $( x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2 )^{2} + 19$ | $C_6\times C_2$ (as 12T2) | $12$ | $12$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |