| $x^{4} + \left(b_{15} \pi^{4} + b_{11} \pi^{3}\right) x^{3} + a_{2} \pi x^{2} + \left(b_{13} \pi^{4} + a_{9} \pi^{3}\right) x + c_{16} \pi^{5} + c_{4} \pi^{2} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $C_4\times C_2$ (show 4), $D_4$ (show 2), $Q_8$ (show 2), $D_4\times C_2$ (show 4), $Q_8:C_2$ (show 8) |
| Hidden Artin slopes: | $[\ ]$ (show 8), $[\ ]^{2}$ (show 12) |
| Indices of inseparability: | $[17,10,4,0]$ |
| Associated inertia: | $[1,1,1]$ |
| Jump Set: | $[1,2,4,8]$ (show 1), $[1,2,4,16]$ (show 10), $[1,11,19,27]$ (show 5), $[1,13,21,29]$ (show 2), $[1,15,23,31]$ (show 1), $[1,16,24,32]$ (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 |
| 2.1.8.24c1.4 |
$x^{8} + 8 x^{5} + 2 x^{4} + 4 x^{2} + 8 x + 2$ |
$Q_8:C_2$ (as 8T11) |
$16$ |
$4$ |
$[2, 3, 4]^{2}$ |
$[1,2,3]^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$[17, 10, 4, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 13, 21, 29]$ |
| 2.1.8.24c1.5 |
$x^{8} + 8 x^{7} + 8 x^{5} + 2 x^{4} + 4 x^{2} + 8 x + 2$ |
$D_4\times C_2$ (as 8T9) |
$16$ |
$4$ |
$[2, 3, 4]^{2}$ |
$[1,2,3]^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$[17, 10, 4, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 13, 21, 29]$ |
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