| $x^{4} + b_{7} \pi^{2} x^{3} + a_{2} \pi x^{2} + a_{5} \pi^{2} x + c_{8} \pi^{3} + c_{4} \pi^{2} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| 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 |
| 2.1.8.16c1.1 |
$x^{8} + 2 x^{6} + 4 x + 2$ |
$C_4\wr C_2$ (as 8T17) |
$32$ |
$4$ |
$[2, 2, \frac{5}{2}]^{4}$ |
$[1,1,\frac{3}{2}]^{4}$ |
$[\ ]^{4}$ |
$[\ ]^{4}$ |
$[9, 6, 6, 0]$ |
$[2, 1]$ |
$z^6 + 1,z + 1$ |
$[1, 3, 9, 17]$ |
| 2.1.8.16c1.2 |
$x^{8} + 2 x^{6} + 4 x^{4} + 4 x + 2$ |
$C_4\wr C_2$ (as 8T17) |
$32$ |
$4$ |
$[2, 2, \frac{5}{2}]^{4}$ |
$[1,1,\frac{3}{2}]^{4}$ |
$[\ ]^{4}$ |
$[\ ]^{4}$ |
$[9, 6, 6, 0]$ |
$[2, 1]$ |
$z^6 + 1,z + 1$ |
$[1, 3, 9, 17]$ |
| 2.1.8.16c1.3 |
$x^{8} + 2 x^{6} + 4 x^{3} + 4 x + 2$ |
$QD_{16}$ (as 8T8) |
$16$ |
$2$ |
$[2, 2, \frac{5}{2}]^{2}$ |
$[1,1,\frac{3}{2}]^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$[9, 6, 6, 0]$ |
$[2, 1]$ |
$z^6 + 1,z + 1$ |
$[1, 3, 9, 17]$ |
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