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 |
| 2.2.8.32c1.12 |
$( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 6$ |
$QD_{16}$ (as 16T12) |
$16$ |
$16$ |
$[2, 2, \frac{5}{2}]^{2}$ |
$[1,1,\frac{3}{2}]^{2}$ |
$[\ ]$ |
$[\ ]$ |
$[9, 6, 6, 0]$ |
$[1, 1]$ |
$z^6 + 1,z + 1$ |
$[1, 3, 9, 17]$ |
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