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.
| 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 |
| 2.1.16.48o2.41 |
$x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 4 x^{4} + 8 x + 6$ |
$C_2\wr C_6$ (as 16T719) |
$384$ |
$2$ |
$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{3}$ |
$[1,1,1,2,2,2,\frac{5}{2}]^{3}$ |
$[2,2,3]^{3}$ |
$[1,1,2]^{3}$ |
$[33, 26, 20, 8, 0]$ |
$[1, 3, 1]$ |
$z^8 + 1,z^6 + z^2 + 1,z + 1$ |
$[1, 2, 4, 8, 32]$ |
| 2.1.16.48o2.45 |
$x^{16} + 4 x^{14} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 4 x^{4} + 8 x + 6$ |
$C_2\wr C_6$ (as 16T719) |
$384$ |
$2$ |
$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{3}$ |
$[1,1,1,2,2,2,\frac{5}{2}]^{3}$ |
$[2,2,3]^{3}$ |
$[1,1,2]^{3}$ |
$[33, 26, 20, 8, 0]$ |
$[1, 3, 1]$ |
$z^8 + 1,z^6 + z^2 + 1,z + 1$ |
$[1, 2, 4, 8, 32]$ |
| 2.1.16.48o2.59 |
$x^{16} + 4 x^{14} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 4 x^{4} + 8 x + 6$ |
$C_2\wr C_6$ (as 16T719) |
$384$ |
$2$ |
$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{3}$ |
$[1,1,1,2,2,2,\frac{5}{2}]^{3}$ |
$[2,2,3]^{3}$ |
$[1,1,2]^{3}$ |
$[33, 26, 20, 8, 0]$ |
$[1, 3, 1]$ |
$z^8 + 1,z^6 + z^2 + 1,z + 1$ |
$[1, 2, 4, 8, 32]$ |
| 2.1.16.48o2.61 |
$x^{16} + 4 x^{14} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 4 x^{4} + 8 x + 6$ |
$C_2\wr C_6$ (as 16T719) |
$384$ |
$2$ |
$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{3}$ |
$[1,1,1,2,2,2,\frac{5}{2}]^{3}$ |
$[2,2,3]^{3}$ |
$[1,1,2]^{3}$ |
$[33, 26, 20, 8, 0]$ |
$[1, 3, 1]$ |
$z^8 + 1,z^6 + z^2 + 1,z + 1$ |
$[1, 2, 4, 8, 32]$ |
