| $x^{4} + \left(b_{31} \pi^{8} + b_{27} \pi^{7} + b_{23} \pi^{6} + b_{19} \pi^{5}\right) x^{3} + a_{2} \pi x^{2} + \left(b_{29} \pi^{8} + b_{25} \pi^{7} + b_{21} \pi^{6} + a_{17} \pi^{5}\right) x + c_{32} \pi^{9} + c_{4} \pi^{2} + \pi$ |
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: | $QD_{16}$ (show 2), $Q_{16}$ (show 2), $D_8:C_2$ (show 3), $D_8:C_2$ (show 2), $C_4\wr C_2$ (show 4), $D_8:C_2$ (show 2), $D_8:C_2$ (show 2), $D_8:C_2$ (show 2), $C_2\times \SD_{16}$ (show 2), $Q_{16}:C_2$ (show 1), $C_4^2:C_2^2$ (show 2), $C_4^2:C_2^2$ (show 4), $C_4^2:C_2^2$ (show 8), $D_4:D_4$ (show 2), $C_2^3.D_4$ (show 4), $C_2\wr C_4$ (show 4), $C_4^2:C_2^2$ (show 2), $C_4^2:D_4$ (show 2), $C_2^4.D_4$ (show 4), $\OD_{16}:D_4$ (show 4), $C_4^2:D_4$ (show 4), $C_4^2:D_4$ (show 4), $C_4^2.D_4$ (show 8), $C_4^2.D_4$ (show 8), $C_2\wr D_4$ (show 8), $C_4^2:D_4$ (show 2), $C_2\wr D_4$ (show 4), $C_4^2:D_4$ (show 4), $C_4^2:D_4$ (show 2), $C_2^5.(C_2\times D_4)$ (show 16), $C_2^4.(C_4\times D_4)$ (show 16), $C_2^6.D_4$ (show 8) (incomplete) |
| Hidden Artin slopes: | $[\ ]$ (show 4), $[\ ]^{2}$ (show 14), $[\frac{7}{2}]^{2}$ (show 18), $[2,\frac{7}{2}]^{2}$ (show 4), $[\frac{7}{2},\frac{17}{4}]^{2}$ (show 40), $[3,\frac{7}{2}]^{2}$ (show 10), $[\frac{7}{2},\frac{17}{4}]$ (show 8), $[3,\frac{7}{2},4,\frac{17}{4}]^{2}$ (show 24), $[2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]^{2}$ (show 16), $[\frac{7}{2}]$ (show 4) (incomplete) |
| Indices of inseparability: | $[49,34,20,8,0]$ |
| Associated inertia: | $[1,1,1,1]$ |
| Jump Set: | $[1,2,4,8,32]$ (show 88), $[1,2,4,32,48]$ (show 54) |
| 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 |
| 2.1.16.64g1.1229 |
$x^{16} + 8 x^{10} + 16 x^{9} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 26$ |
$C_4^2:C_2^2$ (as 16T175) |
$64$ |
$4$ |
$[2, 3, \frac{7}{2}, 4, 5]^{2}$ |
$[1,2,\frac{5}{2},3,4]^{2}$ |
$[\frac{7}{2}]^{2}$ |
$[\frac{5}{2}]^{2}$ |
$[49, 34, 20, 8, 0]$ |
$[1, 1, 1, 1]$ |
$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ |
$[1, 2, 4, 32, 48]$ |
| 2.1.16.64g1.1232 |
$x^{16} + 16 x^{15} + 16 x^{13} + 8 x^{10} + 16 x^{9} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 26$ |
$C_4^2:C_2^2$ (as 16T175) |
$64$ |
$4$ |
$[2, 3, \frac{7}{2}, 4, 5]^{2}$ |
$[1,2,\frac{5}{2},3,4]^{2}$ |
$[\frac{7}{2}]^{2}$ |
$[\frac{5}{2}]^{2}$ |
$[49, 34, 20, 8, 0]$ |
$[1, 1, 1, 1]$ |
$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ |
$[1, 2, 4, 32, 48]$ |
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