| $x^{2} + a_{1} \pi x + c_{2} \pi^{2} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $D_4$ (show 2), $C_2^3: C_4$ (show 1) |
| Hidden Artin slopes: | $[\ ]$ (show 2), $[2]^{2}$ (show 1) |
| Indices of inseparability: | $[3,2,0]$ (show 1), $[3,3,0]$ (show 2) |
| Associated inertia: | $[1]$ (show 2), $[2]$ (show 1) |
| 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 |
| 2.2.4.12a1.1 |
$( x^{2} + x + 1 )^{4} + 2 ( x^{2} + x + 1 )^{3} + 2$ |
$D_4$ (as 8T4) |
$8$ |
$8$ |
$[2, 2]^{2}$ |
$[1,1]^{2}$ |
$[\ ]$ |
$[\ ]$ |
$[3, 3, 0]$ |
$[1]$ |
$z^3 + 1$ |
$[1, 3, 6]$ |
| 2.2.4.12a1.3 |
$( x^{2} + x + 1 )^{4} + 2 ( x^{2} + x + 1 )^{3} + 6$ |
$D_4$ (as 8T4) |
$8$ |
$8$ |
$[2, 2]^{2}$ |
$[1,1]^{2}$ |
$[\ ]$ |
$[\ ]$ |
$[3, 3, 0]$ |
$[1]$ |
$z^3 + 1$ |
$[1, 3, 6]$ |
| 2.2.4.12a6.1 |
$( x^{2} + x + 1 )^{4} + \left(2 x + 2\right) ( x^{2} + x + 1 )^{3} + 2 x ( x^{2} + x + 1 )^{2} + 2$ |
$C_2^3: C_4$ (as 8T21) |
$32$ |
$2$ |
$[2, 2, 2]^{4}$ |
$[1,1,1]^{4}$ |
$[2]^{2}$ |
$[1]^{2}$ |
$[3, 2, 0]$ |
$[2]$ |
$z^3 + t z + t$ |
$[1, 3, 6]$ |
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