Defining polynomial
| $x^{2} + d_{0} \pi$ |
Invariants
| Residue field characteristic: | $5$ |
| Degree: | $2$ |
| Base field: | 5.2.5.12a1.1 |
| Ramification index $e$: | $2$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $1$ |
| Absolute Artin slopes: | $[\frac{3}{2}]$ |
| Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Field count: | $2$ (incomplete) |
| Ambiguity: | $2$ |
| Mass: | $1$ |
| Absolute Mass: | $1/2$ ($3/10$ 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.
| Galois group: | $D_{10}$ (show 1), $C_4\times D_5$ (show 1) (incomplete) |
| Hidden Artin slopes: | not computed (show 1), $[\ ]$ (show 1) (incomplete) |
| Indices of inseparability: | $[4,0]$ |
| Associated inertia: | $[1,1]$ (show 1), $[1,2]$ (show 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 |
|---|---|---|---|---|---|---|---|---|
| 5.2.10.26a2.4 | $( x^{2} + 4 x + 2 )^{10} + 10 ( x^{2} + 4 x + 2 )^{4} + 5$ | $D_{10}$ (as 20T4) | $20$ | $20$ | $[\ ]$ | $[4, 0]$ | $[1, 1]$ | undefined |
| 5.2.10.26a14.1 | $( x^{2} + 4 x + 2 )^{10} + \left(10 x + 20\right) ( x^{2} + 4 x + 2 )^{4} + \left(20 x + 15\right) ( x^{2} + 4 x + 2 )^{2} + \left(5 x + 10\right) ( x^{2} + 4 x + 2 ) + 5 x$ | $C_4\times D_5$ (as 20T6) | $40$ | $4$ | not computed | $[4, 0]$ | $[1, 2]$ | undefined |