| $x^{2} + \left(b_{5} \pi^{3} + a_{3} \pi^{2}\right) x + c_{6} \pi^{4} + \pi$ |
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 |
| 2.1.4.10a1.1 |
$x^{4} + 4 x^{3} + 2$ |
$D_{4}$ (as 4T3) |
$8$ |
$2$ |
$[2, 3, \frac{7}{2}]$ |
$[1,2,\frac{5}{2}]$ |
$[2]$ |
$[1]$ |
$[7, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
| 2.1.4.10a1.2 |
$x^{4} + 4 x^{3} + 8 x^{2} + 2$ |
$D_{4}$ (as 4T3) |
$8$ |
$2$ |
$[2, 3, \frac{7}{2}]$ |
$[1,2,\frac{5}{2}]$ |
$[2]$ |
$[1]$ |
$[7, 4, 0]$ |
$[1, 1]$ |
$z^2 + 1,z + 1$ |
$[1, 3, 7]$ |
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