| $x^{4} + b_{6} \pi^{2} x^{2} + a_{5} \pi^{2} x + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $D_4\times S_4$ |
| Hidden Artin slopes: | $[2]^{2}_{3}$ |
| Indices of inseparability: | $[33,26,20,16,0]$ |
| Associated inertia: | $[1,1,1]$ |
| Jump Set: | $[1,3,7,15,31]$ |
| 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 |
| 2.1.16.48l1.29 |
$x^{16} + 4 x^{14} + 4 x^{10} + 8 x^{5} + 4 x^{4} + 8 x^{3} + 8 x + 2$ |
$D_4\times S_4$ (as 16T421) |
$192$ |
$2$ |
$[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{7}{2}]_{3}^{2}$ |
$[1,\frac{5}{3},\frac{5}{3},2,\frac{5}{2}]_{3}^{2}$ |
$[2]^{2}_{3}$ |
$[1]^{2}_{3}$ |
$[33, 26, 20, 16, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15, 31]$ |
| 2.1.16.48l1.42 |
$x^{16} + 4 x^{10} + 12 x^{8} + 4 x^{4} + 8 x + 2$ |
$D_4\times S_4$ (as 16T421) |
$192$ |
$2$ |
$[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{7}{2}]_{3}^{2}$ |
$[1,\frac{5}{3},\frac{5}{3},2,\frac{5}{2}]_{3}^{2}$ |
$[2]^{2}_{3}$ |
$[1]^{2}_{3}$ |
$[33, 26, 20, 16, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15, 31]$ |
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