| $x^{2} + a_{1} \pi x + c_{2} \pi^{2} + \pi$ |
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.
| Galois group: | $D_4$ (show 1), $D_4\times C_2$ (show 1) (incomplete) |
| Hidden Artin slopes: | $[\ ]$ (show 1), $[\ ]^{2}$ (show 1) (incomplete) |
| Indices of inseparability: | $[17,10,4,0]$ |
| Associated inertia: | $[1,1,1]$ |
| Jump Set: | $[1,2,4,16]$ (show 1), $[1,2,16,24]$ (show 1) |
| 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 |
| 2.1.8.24c1.19 |
$x^{8} + 8 x^{5} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 26$ |
$D_4$ (as 8T4) |
$8$ |
$8$ |
$[2, 3, 4]$ |
$[1,2,3]$ |
$[\ ]$ |
$[\ ]$ |
$[17, 10, 4, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 2, 16, 24]$ |
| 2.1.8.24c1.58 |
$x^{8} + 4 x^{6} + 8 x^{5} + 2 x^{4} + 4 x^{2} + 8 x + 14$ |
$D_4\times C_2$ (as 8T9) |
$16$ |
$4$ |
$[2, 3, 4]^{2}$ |
$[1,2,3]^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$[17, 10, 4, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 2, 4, 16]$ |
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