| $x^{2} + \left(b_{5} \pi^{3} + a_{3} \pi^{2}\right) x + c_{6} \pi^{4} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $QD_{16}$ (show 1), $Q_{16}$ (show 1), $C_4\wr C_2$ (show 1), $C_4^2:C_4$ (show 1), $C_2^4.Q_{16}$ (show 2) |
| Hidden Artin slopes: | $[\ ]$ (show 2), $[2,2,\frac{5}{2}]^{2}$ (show 2), $[\ ]^{2}$ (show 1), $[2]^{2}$ (show 1) |
| Indices of inseparability: | $[9,6,6,0]$ |
| Associated inertia: | $[1,1]$ |
| Jump Set: | $[1,3,9,17]$ |
| 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 |
| 2.2.8.32c1.10 |
$( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 ) + 6$ |
$C_4\wr C_2$ (as 16T42) |
$32$ |
$8$ |
$[2, 2, \frac{5}{2}]^{4}$ |
$[1,1,\frac{3}{2}]^{4}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$[9, 6, 6, 0]$ |
$[1, 1]$ |
$z^6 + 1,z + 1$ |
$[1, 3, 9, 17]$ |
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