These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $(C_4\times C_8).D_4$ (show 4), $C_2^2.C_2\wr C_4$ (show 8), $D_4^2:C_4$ (show 4), $Q_8^2:C_8$ (show 16), $C_4^3.D_4$ (show 16), $(C_4\times \OD_{16}).D_4$ (show 8), $D_4^2:C_8$ (show 16), $Q_8^2:(C_2\times C_4)$ (show 4), $D_4^2:(C_2\times C_4)$ (show 4), $C_2^6.C_2\wr C_4$ (show 64), $C_2^7.C_2\wr C_4$ (show 64), $C_2^7.C_2\wr C_4$ (show 64) (incomplete) |
| Hidden Artin slopes: | not computed (show 96), $[2,2,3,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4},\frac{19}{4}]^{2}$ (show 64), $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4},\frac{19}{4}]^{2}$ (show 32), $[2,2,\frac{7}{2},\frac{9}{2}]^{2}$ (show 16), $[2,2,\frac{7}{2},\frac{7}{2},\frac{9}{2}]$ (show 48), $[2,2,\frac{7}{2},\frac{9}{2}]$ (show 16) (incomplete) |
| Indices of inseparability: | $[24,16,8,0]$ |
| Associated inertia: | $[1,1,1]$ |
| 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 |
| 2.2.8.62a1.8485 |
$( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{6} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + \left(16 x + 24\right) ( x^{2} + x + 1 ) + 8 x + 2$ |
$(C_4\times C_8).D_4$ (as 16T658) |
$256$ |
$2$ |
$[2, 2, 3, \frac{7}{2}, 4, \frac{9}{2}, 5]^{2}$ |
$[1,1,2,\frac{5}{2},3,\frac{7}{2},4]^{2}$ |
$[2,2,\frac{7}{2},\frac{9}{2}]$ |
$[1,1,\frac{5}{2},\frac{7}{2}]$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.2.8.62a1.8486 |
$( x^{2} + x + 1 )^{8} + 16 ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{6} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + \left(16 x + 24\right) ( x^{2} + x + 1 ) + 8 x + 2$ |
$(C_4\times C_8).D_4$ (as 16T658) |
$256$ |
$2$ |
$[2, 2, 3, \frac{7}{2}, 4, \frac{9}{2}, 5]^{2}$ |
$[1,1,2,\frac{5}{2},3,\frac{7}{2},4]^{2}$ |
$[2,2,\frac{7}{2},\frac{9}{2}]$ |
$[1,1,\frac{5}{2},\frac{7}{2}]$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.2.8.62a1.8581 |
$( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{6} + \left(8 x + 28\right) ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + \left(16 x + 24\right) ( x^{2} + x + 1 ) + 8 x + 2$ |
$(C_4\times C_8).D_4$ (as 16T658) |
$256$ |
$2$ |
$[2, 2, 3, \frac{7}{2}, 4, \frac{9}{2}, 5]^{2}$ |
$[1,1,2,\frac{5}{2},3,\frac{7}{2},4]^{2}$ |
$[2,2,\frac{7}{2},\frac{9}{2}]$ |
$[1,1,\frac{5}{2},\frac{7}{2}]$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
| 2.2.8.62a1.8582 |
$( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{6} + \left(8 x + 28\right) ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + \left(16 x + 24\right) ( x^{2} + x + 1 ) + 8 x + 2$ |
$(C_4\times C_8).D_4$ (as 16T658) |
$256$ |
$2$ |
$[2, 2, 3, \frac{7}{2}, 4, \frac{9}{2}, 5]^{2}$ |
$[1,1,2,\frac{5}{2},3,\frac{7}{2},4]^{2}$ |
$[2,2,\frac{7}{2},\frac{9}{2}]$ |
$[1,1,\frac{5}{2},\frac{7}{2}]$ |
$[24, 16, 8, 0]$ |
$[1, 1, 1]$ |
$z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15]$ |
