| $x^{3} + b_{8} \pi^{3} x^{2} + b_{7} \pi^{3} x + c_{9} \pi^{4} + \pi$ |
These invariants are all associated to absolute extensions of $\Q_{ 3 }$ 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 |
| 3.1.6.11a2.1 |
$x^{6} + 6$ |
$D_{6}$ (as 6T3) |
$12$ |
$2$ |
$[\frac{5}{2}]_{2}^{2}$ |
$[\frac{3}{2}]_{2}^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$[6, 0]$ |
$[1, 2]$ |
$z^3 + 2,2 z^2 + 2$ |
undefined |
| 3.1.6.11a2.2 |
$x^{6} + 9 x^{2} + 6$ |
$D_{6}$ (as 6T3) |
$12$ |
$2$ |
$[\frac{5}{2}]_{2}^{2}$ |
$[\frac{3}{2}]_{2}^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$[6, 0]$ |
$[1, 2]$ |
$z^3 + 2,2 z^2 + 2$ |
undefined |
| 3.1.6.11a2.3 |
$x^{6} + 18 x^{2} + 6$ |
$D_{6}$ (as 6T3) |
$12$ |
$2$ |
$[\frac{5}{2}]_{2}^{2}$ |
$[\frac{3}{2}]_{2}^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$[6, 0]$ |
$[1, 2]$ |
$z^3 + 2,2 z^2 + 2$ |
undefined |
| 3.1.6.11a2.4 |
$x^{6} + 9 x + 6$ |
$S_3^2$ (as 6T9) |
$36$ |
$1$ |
$[2, \frac{5}{2}]_{2}^{2}$ |
$[1,\frac{3}{2}]_{2}^{2}$ |
$[2]^{2}$ |
$[1]^{2}$ |
$[6, 0]$ |
$[1, 2]$ |
$z^3 + 2,2 z^2 + 2$ |
undefined |
| 3.1.6.11a2.5 |
$x^{6} + 9 x^{2} + 9 x + 6$ |
$S_3^2$ (as 6T9) |
$36$ |
$1$ |
$[2, \frac{5}{2}]_{2}^{2}$ |
$[1,\frac{3}{2}]_{2}^{2}$ |
$[2]^{2}$ |
$[1]^{2}$ |
$[6, 0]$ |
$[1, 2]$ |
$z^3 + 2,2 z^2 + 2$ |
undefined |
| 3.1.6.11a2.6 |
$x^{6} + 18 x^{2} + 9 x + 6$ |
$S_3^2$ (as 6T9) |
$36$ |
$1$ |
$[2, \frac{5}{2}]_{2}^{2}$ |
$[1,\frac{3}{2}]_{2}^{2}$ |
$[2]^{2}$ |
$[1]^{2}$ |
$[6, 0]$ |
$[1, 2]$ |
$z^3 + 2,2 z^2 + 2$ |
undefined |
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