| $x^{2} + a_{1} \pi x + c_{2} \pi^{2} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $C_4\wr C_2$ (show 1), $C_4^2:C_2^2$ (show 1) |
| Hidden Artin slopes: | $[\frac{7}{2}]^{2}$ (show 1), $[\frac{7}{2}]$ (show 1) |
| Indices of inseparability: | $[49,34,20,8,0]$ |
| Associated inertia: | $[1,1,1,1]$ |
| Jump Set: | $[1,2,4,8,32]$ (show 1), $[1,2,4,32,48]$ (show 1) |
| 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 |
| 2.1.16.64g1.1174 |
$x^{16} + 16 x^{15} + 16 x^{13} + 16 x^{11} + 8 x^{10} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 10$ |
$C_4\wr C_2$ (as 16T42) |
$32$ |
$8$ |
$[2, 3, \frac{7}{2}, 4, 5]$ |
$[1,2,\frac{5}{2},3,4]$ |
$[\frac{7}{2}]$ |
$[\frac{5}{2}]$ |
$[49, 34, 20, 8, 0]$ |
$[1, 1, 1, 1]$ |
$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ |
$[1, 2, 4, 32, 48]$ |
| 2.1.16.64g1.4480 |
$x^{16} + 4 x^{12} + 16 x^{11} + 8 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{7} + 4 x^{4} + 8 x^{2} + 16 x + 46$ |
$C_4^2:C_2^2$ (as 16T111) |
$64$ |
$8$ |
$[2, 3, \frac{7}{2}, 4, 5]^{2}$ |
$[1,2,\frac{5}{2},3,4]^{2}$ |
$[\frac{7}{2}]^{2}$ |
$[\frac{5}{2}]^{2}$ |
$[49, 34, 20, 8, 0]$ |
$[1, 1, 1, 1]$ |
$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ |
$[1, 2, 4, 8, 32]$ |
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