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 $\Q_{ 2 }$ within this relative family, not the relative extension.
| Label |
Polynomial $/ \Q_p$ |
Galois group $/ \Q_p$ |
Galois degree $/ \Q_p$ |
$\#\Aut(K/\Q_p)$ |
Artin slope content $/ \Q_p$ |
Swan slope content $/ \Q_p$ |
Hidden Artin slopes $/ \Q_p$ |
Hidden Swan slopes $/ \Q_p$ |
Ind. of Insep. $/ \Q_p$ |
Assoc. Inertia $/ \Q_p$ |
Resid. Poly |
Jump Set |
| 2.2.8.48c1.5 |
$( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 16 x + 2$ |
$Q_8 : C_2$ (as 16T11) |
$16$ |
$16$ |
$[2, 3, 4]^{2}$ |
$[1,2,3]^{2}$ |
$[\ ]$ |
$[\ ]$ |
$[17, 10, 4, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 15, 23, 31]$ |
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