Defining polynomial
| $x^{3} + 7d_{0}$ |
Invariants
| Residue field characteristic: | $7$ |
| Degree: | $3$ |
| Base field: | 7.4.1.0a1.1 |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $2$ |
| Absolute Artin slopes: | $[\ ]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $3$ (complete) |
| Ambiguity: | $3$ |
| Mass: | $1$ |
| Absolute Mass: | $1/4$ |
Varying
These invariants are all associated to absolute extensions of $\Q_{ 7 }$ within this relative family, not the relative extension.
| Galois group: | $C_{12}$ |
| Hidden Artin slopes: | $[\ ]$ |
| Indices of inseparability: | $[0]$ |
| Associated inertia: | $[1]$ |
| Jump Set: | undefined |
Fields
Showing all 3
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 |
|---|---|---|---|---|---|---|---|---|
| 7.4.3.8a1.1 | $( x^{4} + 5 x^{2} + 4 x + 3 )^{3} + 7 x^{2}$ | $C_{12}$ (as 12T1) | $12$ | $12$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 7.4.3.8a1.2 | $( x^{4} + 5 x^{2} + 4 x + 3 )^{3} + 7 x$ | $C_{12}$ (as 12T1) | $12$ | $12$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |
| 7.4.3.8a1.3 | $( x^{4} + 5 x^{2} + 4 x + 3 )^{3} + 7$ | $C_{12}$ (as 12T1) | $12$ | $12$ | $[\ ]$ | $[0]$ | $[1]$ | undefined |