| $x^{2} + \left(b_{17} \pi^{9} + b_{15} \pi^{8} + b_{13} \pi^{7} + b_{11} \pi^{6} + a_{9} \pi^{5}\right) x + c_{18} \pi^{10} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $C_4^2:C_4$ (show 2), $C_2^4.D_4$ (show 2), $C_2^4.(C_4\times D_4)$ (show 4), $C_2^5.(C_2\times D_4)$ (show 4), $C_2^6.D_4$ (show 8) |
| Hidden Artin slopes: | $[2,3,\frac{7}{2},4]^{2}$ (show 12), $[2,\frac{7}{2}]^{2}$ (show 2), $[2,\frac{7}{2}]$ (show 2), $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ (show 4) |
| Indices of inseparability: | $[51,42,32,16,0]$ |
| Associated inertia: | $[1,1,1,1]$ |
| Jump Set: | $[1,3,7,15,31]$ |
| 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 |
| 2.1.16.66j1.529 |
$x^{16} + 8 x^{14} + 8 x^{10} + 8 x^{4} + 16 x^{3} + 10$ |
$C_2^4.D_4$ (as 16T297) |
$128$ |
$2$ |
$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ |
$[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ |
$[2,\frac{7}{2}]^{2}$ |
$[1,\frac{5}{2}]^{2}$ |
$[51, 42, 32, 16, 0]$ |
$[1, 1, 1, 1]$ |
$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15, 31]$ |
| 2.1.16.66j1.530 |
$x^{16} + 8 x^{14} + 16 x^{12} + 8 x^{10} + 8 x^{4} + 16 x^{3} + 10$ |
$C_2^4.D_4$ (as 16T297) |
$128$ |
$2$ |
$[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ |
$[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ |
$[2,\frac{7}{2}]^{2}$ |
$[1,\frac{5}{2}]^{2}$ |
$[51, 42, 32, 16, 0]$ |
$[1, 1, 1, 1]$ |
$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ |
$[1, 3, 7, 15, 31]$ |
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