| $x^{2} + a_{1} \pi x + c_{2} \pi^{2} + \pi$ |
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: | $C_2\wr C_2^2$ (show 1), $C_2\wr C_2^2$ (show 1) (incomplete) |
| Hidden Artin slopes: | $[\frac{7}{2}]^{2}$ (incomplete) |
| Indices of inseparability: | $[43,34,20,8,0]$ |
| Associated inertia: | $[1,1,1,1]$ |
| Jump Set: | $[1,2,4,8,32]$ (show 1), $[1,11,29,45,61]$ (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 |
| 2.1.16.58n1.84 |
$x^{16} + 8 x^{14} + 8 x^{13} + 8 x^{11} + 2 x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{3} + 8 x^{2} + 2$ |
$C_2\wr C_2^2$ (as 16T129) |
$64$ |
$4$ |
$[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ |
$[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ |
$[\frac{7}{2}]^{2}$ |
$[\frac{5}{2}]^{2}$ |
$[43, 34, 20, 8, 0]$ |
$[1, 1, 1, 1]$ |
$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ |
$[1, 11, 29, 45, 61]$ |
| 2.1.16.58n1.628 |
$x^{16} + 8 x^{15} + 8 x^{13} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 16 x + 6$ |
$C_2\wr C_2^2$ (as 16T149) |
$64$ |
$8$ |
$[2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ |
$[1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$ |
$[\frac{7}{2}]^{2}$ |
$[\frac{5}{2}]^{2}$ |
$[43, 34, 20, 8, 0]$ |
$[1, 1, 1, 1]$ |
$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$ |
$[1, 2, 4, 8, 32]$ |
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