| $x^{4} + a_{3} \pi x^{3} + b_{2} \pi x^{2} + c_{4} \pi^{2} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $C_2^4:C_4$ (show 2), $D_4\times A_4$ (show 1) |
| Hidden Artin slopes: | $[\ ]^{6}$ (show 1), $[\ ]^{4}$ (show 2) |
| Indices of inseparability: | $[23,14,12,8,0]$ (show 2), $[23,14,14,14,0]$ (show 1) |
| Associated inertia: | $[3,1]$ (show 1), $[4,1]$ (show 2) |
| Jump Set: | $[1,2,7,14,32]$ (show 2), $[1,3,7,14,32]$ (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 |
| 2.1.16.38c1.21 |
$x^{16} + 2 x^{14} + 4 x^{7} + 6$ |
$D_4\times A_4$ (as 16T179) |
$96$ |
$2$ |
$[2, 2, 2, 3]^{6}$ |
$[1,1,1,2]^{6}$ |
$[\ ]^{6}$ |
$[\ ]^{6}$ |
$[23, 14, 14, 14, 0]$ |
$[3, 1]$ |
$z^{14} + 1,z + 1$ |
$[1, 3, 7, 14, 32]$ |
| 2.1.16.38c4.25 |
$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{13} + 2 x^{12} + 2 x^{8} + 4 x^{7} + 6$ |
$C_2^4:C_4$ (as 16T76) |
$64$ |
$4$ |
$[2, 2, 2, 3]^{4}$ |
$[1,1,1,2]^{4}$ |
$[\ ]^{4}$ |
$[\ ]^{4}$ |
$[23, 14, 12, 8, 0]$ |
$[4, 1]$ |
$z^{14} + z^6 + z^2 + 1,z + 1$ |
$[1, 2, 7, 14, 32]$ |
| 2.1.16.38c4.26 |
$x^{16} + 4 x^{15} + 2 x^{14} + 4 x^{13} + 2 x^{12} + 2 x^{8} + 4 x^{7} + 14$ |
$C_2^4:C_4$ (as 16T76) |
$64$ |
$4$ |
$[2, 2, 2, 3]^{4}$ |
$[1,1,1,2]^{4}$ |
$[\ ]^{4}$ |
$[\ ]^{4}$ |
$[23, 14, 12, 8, 0]$ |
$[4, 1]$ |
$z^{14} + z^6 + z^2 + 1,z + 1$ |
$[1, 2, 7, 14, 32]$ |
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