| $x^{6} + a_{5} \pi x^{5} + c_{10} \pi^{2} x^{4} + b_{9} \pi^{2} x^{3} + b_{7} \pi^{2} x + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $C_2 \times S_4$ (show 2), $C_2 \times S_4$ (show 2), $C_2^2\times S_4$ (show 4) |
| Hidden Artin slopes: | $[\frac{8}{3}]^{2}$ (show 4), $[2,\frac{8}{3}]^{2}$ (show 4) |
| Indices of inseparability: | $[17,10,0]$ |
| Associated inertia: | $[2,1,1]$ |
| Jump Set: | $[3,9,21]$ |
| 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 |
| 2.1.12.28c1.7 |
$x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 4 x^{5} + 2$ |
$C_2 \times S_4$ (as 12T24) |
$48$ |
$4$ |
$[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ |
$[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ |
$[\frac{8}{3}]^{2}$ |
$[\frac{5}{3}]^{2}$ |
$[17, 10, 0]$ |
$[2, 1, 1]$ |
$z^8 + z^4 + 1,z^2 + 1,z + 1$ |
$[3, 9, 21]$ |
| 2.1.12.28c1.27 |
$x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{6} + 4 x^{5} + 2$ |
$C_2 \times S_4$ (as 12T24) |
$48$ |
$4$ |
$[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ |
$[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ |
$[\frac{8}{3}]^{2}$ |
$[\frac{5}{3}]^{2}$ |
$[17, 10, 0]$ |
$[2, 1, 1]$ |
$z^8 + z^4 + 1,z^2 + 1,z + 1$ |
$[3, 9, 21]$ |
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