| $x^{4} + a_{3} \pi x^{3} + b_{2} \pi x^{2} + c_{4} \pi^{2} + \pi$ |
The following invariants arise for fields within the LMFDB; since not all fields in this family are stored, it may be incomplete.
These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.
| Galois group: | $C_4:D_4$ (show 1), $C_2^2\wr C_2$ (show 1), $D_4\times A_4$ (show 1) (incomplete) |
| Hidden Artin slopes: | $[\ ]^{2}$ (show 2), $[2]^{3}$ (show 1) (incomplete) |
| Indices of inseparability: | $[31,22,12,8,0]$ (show 1), $[31,22,12,12,0]$ (show 2) |
| Associated inertia: | $[2,1,1]$ (show 2), $[3,1,1]$ (show 1) |
| Jump Set: | $[1,2,7,15,31]$ (show 1), $[1,3,6,12,32]$ (show 1), $[1,3,15,32,48]$ (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 |
| 2.1.16.46k1.36 |
$x^{16} + 4 x^{15} + 4 x^{14} + 2 x^{12} + 8 x^{7} + 4 x^{6} + 10$ |
$C_2^2\wr C_2$ (as 16T39) |
$32$ |
$8$ |
$[2, 2, 3, \frac{7}{2}]^{2}$ |
$[1,1,2,\frac{5}{2}]^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$[31, 22, 12, 12, 0]$ |
$[2, 1, 1]$ |
$z^{12} + 1,z^2 + 1,z + 1$ |
$[1, 3, 15, 32, 48]$ |
| 2.1.16.46k1.93 |
$x^{16} + 4 x^{15} + 2 x^{12} + 4 x^{6} + 6$ |
$C_4:D_4$ (as 16T34) |
$32$ |
$4$ |
$[2, 2, 3, \frac{7}{2}]^{2}$ |
$[1,1,2,\frac{5}{2}]^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$[31, 22, 12, 12, 0]$ |
$[2, 1, 1]$ |
$z^{12} + 1,z^2 + 1,z + 1$ |
$[1, 3, 6, 12, 32]$ |
| 2.1.16.46k2.39 |
$x^{16} + 4 x^{15} + 2 x^{12} + 2 x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{3} + 8 x + 10$ |
$D_4\times A_4$ (as 16T179) |
$96$ |
$2$ |
$[2, 2, 2, 3, \frac{7}{2}]^{3}$ |
$[1,1,1,2,\frac{5}{2}]^{3}$ |
$[2]^{3}$ |
$[1]^{3}$ |
$[31, 22, 12, 8, 0]$ |
$[3, 1, 1]$ |
$z^{12} + z^4 + 1,z^2 + 1,z + 1$ |
$[1, 2, 7, 15, 31]$ |
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