| $x^{2} + \left(b_{5} \pi^{3} + a_{3} \pi^{2}\right) x + c_{6} \pi^{4} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $D_4 \times C_3$ (show 2), $D_4\times A_4$ (show 2), $C_2^3:A_4$ (show 2), $C_2^4:A_4$ (show 6), $C_2\wr C_6$ (show 8) |
| Hidden Artin slopes: | $[2,2,2,\frac{7}{2},\frac{7}{2}]$ (show 8), $[2,2,\frac{7}{2}]^{2}$ (show 2), $[2,2,2]$ (show 2), $[2,2,2,\frac{7}{2}]$ (show 4), $[2]$ (show 2), $[2,2,\frac{7}{2}]$ (show 2) |
| Indices of inseparability: | $[7,4,0]$ |
| Associated inertia: | $[1,1]$ |
| Jump Set: | $[1,3,7]$ |
| 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 |
| 2.3.4.30a1.31 |
$( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x ( x^{3} + x + 1 ) + 10$ |
$D_4\times A_4$ (as 12T51) |
$96$ |
$2$ |
$[2, 2, 2, 3, \frac{7}{2}]^{3}$ |
$[1,1,1,2,\frac{5}{2}]^{3}$ |
$[2,2,2]$ |
$[1,1,1]$ |
$[7, 4, 0]$ |
$[1, 1]$ |
$z^2 + (t + 1),(t + 1) z + 1$ |
$[1, 3, 7]$ |
| 2.3.4.30a1.32 |
$( x^{3} + x + 1 )^{4} + 4 ( x^{3} + x + 1 )^{3} + 12 ( x^{3} + x + 1 )^{2} + 8 x ( x^{3} + x + 1 ) + 10$ |
$D_4\times A_4$ (as 12T51) |
$96$ |
$2$ |
$[2, 2, 2, 3, \frac{7}{2}]^{3}$ |
$[1,1,1,2,\frac{5}{2}]^{3}$ |
$[2,2,2]$ |
$[1,1,1]$ |
$[7, 4, 0]$ |
$[1, 1]$ |
$z^2 + (t + 1),(t + 1) z + 1$ |
$[1, 3, 7]$ |
Download
displayed columns for
results
