| $x^{2} + a_{1} \pi x + c_{2} \pi^{2} + \pi$ |
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: | $C_8: C_2$ (show 1), $C_2 \times (C_8:C_2)$ (show 1) (incomplete) |
| Hidden Artin slopes: | $[\ ]$ (show 1), $[\ ]^{2}$ (show 1) (incomplete) |
| Indices of inseparability: | $[49,34,20,8,0]$ |
| Associated inertia: | $[1,1,1,1]$ |
| Jump Set: | $[1,2,4,8,32]$ (show 1), $[1,27,43,59,75]$ (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 |
| 2.1.16.64g1.2 |
$x^{16} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 16 x + 34$ |
$C_8: C_2$ (as 16T6) |
$16$ |
$16$ |
$[2, 3, 4, 5]$ |
$[1,2,3,4]$ |
$[\ ]$ |
$[\ ]$ |
$[49, 34, 20, 8, 0]$ |
$[1, 1, 1, 1]$ |
$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ |
$[1, 27, 43, 59, 75]$ |
| 2.1.16.64g1.4792 |
$x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 16 x^{11} + 16 x^{9} + 2 x^{8} + 16 x^{7} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 30$ |
$C_2 \times (C_8:C_2)$ (as 16T15) |
$32$ |
$8$ |
$[2, 3, 4, 5]^{2}$ |
$[1,2,3,4]^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$[49, 34, 20, 8, 0]$ |
$[1, 1, 1, 1]$ |
$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ |
$[1, 2, 4, 8, 32]$ |
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