Defining polynomial
| $x^{2} + d_{0} \pi$ |
Invariants
| Residue field characteristic: | $5$ |
| Degree: | $2$ |
| Base field: | 5.1.10.19a2.11 |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $1$ |
| Absolute Artin slopes: | $[\frac{9}{4}]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $2$ (incomplete) |
| Ambiguity: | $2$ |
| Mass: | $1$ |
| Absolute Mass: | $1$ ($3/5$ currently in the LMFDB) |
Varying
The following invariants arise for fields within the LMFDB; since not all fields in this family are stored, it may be incomplete.
These invariants are all associated to absolute extensions of $\Q_{ 5 }$ within this relative family, not the relative extension.
| Indices of inseparability: | $[20,0]$ |
| Associated inertia: | $[1,1]$ (show 1), $[1,2]$ (show 1) |
| Jump Set: | undefined (show 1), $[1,21]$ (show 1) |
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 |
|---|---|---|---|---|---|---|---|---|
| 5.1.20.39a1.101 | $x^{20} + 50 x^{2} + 5$ | not computed | $ $not computed$ $ | $10$ | not computed | $[20, 0]$ | $[1, 1]$ | $[1, 21]$ |
| 5.1.20.39a4.12 | $x^{20} + 25 x^{4} + 25 x^{2} + 20$ | not computed | $ $not computed$ $ | $2$ | not computed | $[20, 0]$ | $[1, 2]$ | undefined |