| $x^{2} + \left(b_{9} \pi^{5} + b_{7} \pi^{4} + a_{5} \pi^{3}\right) x + c_{10} \pi^{6} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $C_4\times A_4$ (show 2), $C_2^2\times A_4$ (show 2), $D_4\times A_4$ (show 2), $D_4.A_4$ (show 8), $(C_2^2\times C_4):A_4$ (show 2), $C_2\wr A_4$ (show 2), $C_2^4:(C_2\times A_4)$ (show 10), $C_2^6.(C_4\times A_4)$ (show 24) (incomplete) |
| Hidden Artin slopes: | $[2,2,3]^{3}$ (show 8), not computed (show 24), $[2,2,2]^{3}$ (show 2), $[\ ]^{3}$ (show 4), $[3]^{3}$ (show 8), $[2]^{3}$ (show 2), $[2,2]^{3}$ (show 4) (incomplete) |
| Indices of inseparability: | $[11,6,4,0]$ |
| Associated inertia: | $[3,1]$ |
| Jump Set: | $[1,2,7,15]$ |
| 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 |
| 2.2.8.36b6.3 |
$( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 2$ |
$D_4\times A_4$ (as 16T179) |
$96$ |
$2$ |
$[2, 2, 2, 3]^{6}$ |
$[1,1,1,2]^{6}$ |
$[2]^{3}$ |
$[1]^{3}$ |
$[11, 6, 4, 0]$ |
$[3, 1]$ |
$z^6 + z^2 + 1,z + 1$ |
$[1, 2, 7, 15]$ |
| 2.2.8.36b6.4 |
$( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 2$ |
$D_4\times A_4$ (as 16T179) |
$96$ |
$2$ |
$[2, 2, 2, 3]^{6}$ |
$[1,1,1,2]^{6}$ |
$[2]^{3}$ |
$[1]^{3}$ |
$[11, 6, 4, 0]$ |
$[3, 1]$ |
$z^6 + z^2 + 1,z + 1$ |
$[1, 2, 7, 15]$ |
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