| $x^{2} + a_{1} \pi x + c_{2} \pi^{2} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $C_2^2 : C_4$ (show 2), $C_2^4:C_4$ (show 1) |
| Hidden Artin slopes: | $[\ ]$ (show 2), $[2]^{2}$ (show 1) |
| Indices of inseparability: | $[11,6,4,0]$ (show 1), $[11,6,6,0]$ (show 2) |
| Associated inertia: | $[1,1]$ (show 2), $[2,1]$ (show 1) |
| Jump Set: | $[1,3,16,24]$ |
| 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 |
| 2.2.8.36b1.2 |
$( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 2$ |
$C_2^2 : C_4$ (as 16T10) |
$16$ |
$16$ |
$[2, 2, 3]^{2}$ |
$[1,1,2]^{2}$ |
$[\ ]$ |
$[\ ]$ |
$[11, 6, 6, 0]$ |
$[1, 1]$ |
$z^6 + 1,z + 1$ |
$[1, 3, 16, 24]$ |
| 2.2.8.36b1.28 |
$( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{3} + 8 x + 6$ |
$C_2^2 : C_4$ (as 16T10) |
$16$ |
$16$ |
$[2, 2, 3]^{2}$ |
$[1,1,2]^{2}$ |
$[\ ]$ |
$[\ ]$ |
$[11, 6, 6, 0]$ |
$[1, 1]$ |
$z^6 + 1,z + 1$ |
$[1, 3, 16, 24]$ |
| 2.2.8.36b16.18 |
$( x^{2} + x + 1 )^{8} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{7} + \left(2 x + 2\right) ( x^{2} + x + 1 )^{6} + \left(4 x + 6\right) ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{3} + 8 x + 2$ |
$C_2^4:C_4$ (as 16T78) |
$64$ |
$4$ |
$[2, 2, 2, 3]^{4}$ |
$[1,1,1,2]^{4}$ |
$[2]^{2}$ |
$[1]^{2}$ |
$[11, 6, 4, 0]$ |
$[2, 1]$ |
$z^6 + t z^2 + t,t z + (t + 1)$ |
$[1, 3, 16, 24]$ |
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