| $x^{2} + a_{1} \pi x + c_{2} \pi^{2} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $C_4\times C_2^2$ (show 1), $C_8\times C_2$ (show 1), $C_2 \times (C_8:C_2)$ (show 1), $C_2 \times (C_2^2:C_4)$ (show 1), $C_2^4:C_4$ (show 1), $\OD_{16}:C_2^2$ (show 1), $C_2^5:C_4$ (show 4) |
| Hidden Artin slopes: | $[2,2,2]$ (show 4), $[\ ]$ (show 2), $[2]$ (show 2), $[2,2]$ (show 2) |
| Indices of inseparability: | $[5,2,0]$ |
| Associated inertia: | $[1,1]$ |
| Jump Set: | $[1,2,4]$ (show 1), $[1,2,8]$ (show 1), $[1,3,7]$ (show 8) |
| 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 |
| 2.4.4.32b25.4 |
$( x^{4} + x + 1 )^{4} + \left(4 x^{3} + 4 x^{2} + 2\right) ( x^{4} + x + 1 )^{3} + \left(2 x^{2} + 2 x\right) ( x^{4} + x + 1 )^{2} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 ) + 2$ |
$C_2 \times (C_2^2:C_4)$ (as 16T21) |
$32$ |
$8$ |
$[2, 2, 3]^{4}$ |
$[1,1,2]^{4}$ |
$[2]$ |
$[1]$ |
$[5, 2, 0]$ |
$[1, 1]$ |
$z^2 + (t^2 + t),(t^2 + t) z + (t^2 + t + 1)$ |
$[1, 3, 7]$ |
| 2.4.4.32b25.29 |
$( x^{4} + x + 1 )^{4} + \left(4 x^{3} + 4 x^{2} + 4 x + 6\right) ( x^{4} + x + 1 )^{3} + \left(2 x^{2} + 2 x\right) ( x^{4} + x + 1 )^{2} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 ) + 4 x + 2$ |
$C_2 \times (C_8:C_2)$ (as 16T15) |
$32$ |
$8$ |
$[2, 2, 3]^{4}$ |
$[1,1,2]^{4}$ |
$[2]$ |
$[1]$ |
$[5, 2, 0]$ |
$[1, 1]$ |
$z^2 + (t^2 + t),(t^2 + t) z + (t^2 + t + 1)$ |
$[1, 3, 7]$ |
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