| $x^{4} + \left(b_{11} \pi^{3} + a_{7} \pi^{2}\right) x^{3} + a_{2} \pi x^{2} + b_{9} \pi^{3} x + c_{12} \pi^{4} + c_{4} \pi^{2} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $C_4:C_4$ (show 2), $C_2^2 : C_4$ (show 2), $C_4^2:C_2$ (show 1), $C_4 \times D_4$ (show 4), $C_2 \times (C_2^2:C_4)$ (show 1) |
| Hidden Artin slopes: | $[\ ]$ (show 4), $[\ ]^{2}$ (show 6) |
| Indices of inseparability: | $[39,30,20,8,0]$ |
| Associated inertia: | $[1,1,1,1]$ |
| Jump Set: | $[1,2,4,8,32]$ (show 5), $[1,7,32,48,64]$ (show 5) |
| 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 |
| 2.1.16.54o1.318 |
$x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 2 x^{8} + 8 x^{7} + 4 x^{4} + 8 x^{2} + 14$ |
$C_2 \times (C_2^2:C_4)$ (as 16T21) |
$32$ |
$8$ |
$[2, 3, \frac{7}{2}, 4]^{2}$ |
$[1,2,\frac{5}{2},3]^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$[39, 30, 20, 8, 0]$ |
$[1, 1, 1, 1]$ |
$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ |
$[1, 2, 4, 8, 32]$ |
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