| $x^{3} + a_{5} \pi^{2} x^{2} + b_{7} \pi^{3} x + \pi$ |
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 |
| 3.1.6.10a1.4 |
$x^{6} + 3 x^{5} + 6$ |
$C_3^2:D_4$ (as 6T13) |
$72$ |
$1$ |
$[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$ |
$[\frac{5}{4},\frac{5}{4}]_{4}^{2}$ |
$[\frac{9}{4}]^{2}_{2}$ |
$[\frac{5}{4}]^{2}_{2}$ |
$[5, 0]$ |
$[1, 1]$ |
$z^3 + 2,2 z + 2$ |
undefined |
| 3.1.6.10a1.5 |
$x^{6} + 3 x^{5} + 9 x + 6$ |
$C_3^2:D_4$ (as 6T13) |
$72$ |
$1$ |
$[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$ |
$[\frac{5}{4},\frac{5}{4}]_{4}^{2}$ |
$[\frac{9}{4}]^{2}_{2}$ |
$[\frac{5}{4}]^{2}_{2}$ |
$[5, 0]$ |
$[1, 1]$ |
$z^3 + 2,2 z + 2$ |
undefined |
| 3.1.6.10a1.6 |
$x^{6} + 6 x^{5} + 9 x + 6$ |
$C_3^2:D_4$ (as 6T13) |
$72$ |
$1$ |
$[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$ |
$[\frac{5}{4},\frac{5}{4}]_{4}^{2}$ |
$[\frac{9}{4}]^{2}_{2}$ |
$[\frac{5}{4}]^{2}_{2}$ |
$[5, 0]$ |
$[1, 1]$ |
$z^3 + 2,2 z + 1$ |
undefined |
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