| $x^{4} + a_{7} \pi^{2} x^{3} + b_{6} \pi^{2} x^{2} + b_{9} \pi^{3} x + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $V_4^2:(S_3\times C_2)$ |
| Hidden Artin slopes: | $[\frac{4}{3},\frac{4}{3}]^{2}_{3}$ |
| Indices of inseparability: | $[15,14,8,0]$ (show 1), $[15,15,8,0]$ (show 1) |
| Associated inertia: | $[1,1]$ |
| Jump Set: | $[1,3,7,15]$ |
| 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 |
| 2.1.8.22c1.1 |
$x^{8} + 4 x^{7} + 2$ |
$V_4^2:(S_3\times C_2)$ (as 8T41) |
$192$ |
$1$ |
$[\frac{4}{3}, \frac{4}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ |
$[\frac{1}{3},\frac{1}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ |
$[\frac{4}{3},\frac{4}{3}]^{2}_{3}$ |
$[\frac{1}{3},\frac{1}{3}]^{2}_{3}$ |
$[15, 15, 8, 0]$ |
$[1, 1]$ |
$z^4 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.1.8.22c1.7 |
$x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 2$ |
$V_4^2:(S_3\times C_2)$ (as 8T41) |
$192$ |
$1$ |
$[\frac{4}{3}, \frac{4}{3}, 3, \frac{19}{6}, \frac{19}{6}]_{3}^{2}$ |
$[\frac{1}{3},\frac{1}{3},2,\frac{13}{6},\frac{13}{6}]_{3}^{2}$ |
$[\frac{4}{3},\frac{4}{3}]^{2}_{3}$ |
$[\frac{1}{3},\frac{1}{3}]^{2}_{3}$ |
$[15, 14, 8, 0]$ |
$[1, 1]$ |
$z^4 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
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