| $x^{2} + a_{1} \pi x + c_{2} \pi^{2} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $A_4 \times C_2$ (show 1), $D_4 \times C_3$ (show 1), $C_2^2 \times A_4$ (show 1), $D_4\times A_4$ (show 1) |
| Hidden Artin slopes: | $[2,2]^{2}$ (show 1), $[2]$ (show 1), $[2]^{2}$ (show 1), $[\ ]^{2}$ (show 1) |
| Indices of inseparability: | $[3,2,0]$ (show 3), $[3,3,0]$ (show 1) |
| Associated inertia: | $[1]$ (show 2), $[2]$ (show 2) |
| Jump Set: | $[1,3,6]$ |
| 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 |
| 2.3.4.18a1.1 |
$( x^{3} + x + 1 )^{4} + 2 ( x^{3} + x + 1 )^{3} + 2$ |
$D_4 \times C_3$ (as 12T14) |
$24$ |
$6$ |
$[2, 2]^{6}$ |
$[1,1]^{6}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$[3, 3, 0]$ |
$[2]$ |
$z^3 + (t^2 + t)$ |
$[1, 3, 6]$ |
| 2.3.4.18a13.1 |
$( x^{3} + x + 1 )^{4} + \left(2 x^{2} + 2\right) ( x^{3} + x + 1 )^{3} + 2 x ( x^{3} + x + 1 )^{2} + 2$ |
$C_2^2 \times A_4$ (as 12T25) |
$48$ |
$4$ |
$[2, 2, 2]^{6}$ |
$[1,1,1]^{6}$ |
$[2]^{2}$ |
$[1]^{2}$ |
$[3, 2, 0]$ |
$[1]$ |
$z^3 + (t^2 + 1) z + t^2$ |
$[1, 3, 6]$ |
| 2.3.4.18a13.2 |
$( x^{3} + x + 1 )^{4} + \left(2 x^{2} + 2\right) ( x^{3} + x + 1 )^{3} + 2 x ( x^{3} + x + 1 )^{2} + 4 x + 2$ |
$A_4 \times C_2$ (as 12T7) |
$24$ |
$4$ |
$[2, 2, 2]^{3}$ |
$[1,1,1]^{3}$ |
$[2]$ |
$[1]$ |
$[3, 2, 0]$ |
$[1]$ |
$z^3 + (t^2 + 1) z + t^2$ |
$[1, 3, 6]$ |
| 2.3.4.18a19.1 |
$( x^{3} + x + 1 )^{4} + 2 x^{2} ( x^{3} + x + 1 )^{3} + \left(2 x + 2\right) ( x^{3} + x + 1 )^{2} + 2$ |
$D_4\times A_4$ (as 12T51) |
$96$ |
$2$ |
$[2, 2, 2, 2]^{6}$ |
$[1,1,1,1]^{6}$ |
$[2,2]^{2}$ |
$[1,1]^{2}$ |
$[3, 2, 0]$ |
$[2]$ |
$z^3 + t z + t$ |
$[1, 3, 6]$ |
Download
displayed columns for
results
