Defining polynomial
| $x^{2} + d_{0} \pi$ |
Invariants
| Residue field characteristic: | $7$ |
| Degree: | $2$ |
| Base field: | 7.1.7.10a1.2 |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $1$ |
| Absolute Artin slopes: | $[\frac{5}{3}]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $2$ (complete) |
| Ambiguity: | $2$ |
| Mass: | $1$ |
| Absolute Mass: | $1$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 7 }$ within this relative family, not the relative extension.
| Galois group: | $F_7 \times C_2$ |
| Hidden Artin slopes: | $[\ ]^{2}_{3}$ |
| Indices of inseparability: | $[8,0]$ |
| Associated inertia: | $[1,2]$ |
| 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 |
|---|---|---|---|---|---|---|---|---|
| 7.1.14.21a1.5 | $x^{14} + 35 x^{8} + 7$ | $F_7 \times C_2$ (as 14T7) | $84$ | $2$ | $[\ ]^{2}_{3}$ | $[8, 0]$ | $[1, 2]$ | undefined |
| 7.1.14.21a1.17 | $x^{14} + 14 x^{8} + 21$ | $F_7 \times C_2$ (as 14T7) | $84$ | $2$ | $[\ ]^{2}_{3}$ | $[8, 0]$ | $[1, 2]$ | undefined |