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 |
| 3.1.12.19a1.8 |
$x^{12} + 3 x^{11} + 6 x^{8} + 6$ |
$C_3^2:D_{12}$ (as 12T118) |
$216$ |
$1$ |
$[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$ |
$[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$ |
$[\frac{5}{4},\frac{5}{4}]^{2}$ |
$[\frac{1}{4},\frac{1}{4}]^{2}$ |
$[8, 0]$ |
$[2, 2]$ |
$z^9 + z^6 + 1,z^2 + 1$ |
undefined |
| 3.1.12.19a2.22 |
$x^{12} + 3 x^{11} + 3 x^{8} + 6$ |
$S_3^2:C_6$ (as 12T121) |
$216$ |
$3$ |
$[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$ |
$[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$ |
$[\frac{5}{4},\frac{5}{4}]^{2}$ |
$[\frac{1}{4},\frac{1}{4}]^{2}$ |
$[8, 0]$ |
$[2, 1]$ |
$z^9 + z^6 + 1,z^2 + 2$ |
undefined |
| 3.1.12.19a2.23 |
$x^{12} + 3 x^{11} + 3 x^{8} + 15$ |
$S_3^2:C_6$ (as 12T121) |
$216$ |
$3$ |
$[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$ |
$[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$ |
$[\frac{5}{4},\frac{5}{4}]^{2}$ |
$[\frac{1}{4},\frac{1}{4}]^{2}$ |
$[8, 0]$ |
$[2, 1]$ |
$z^9 + z^6 + 1,z^2 + 2$ |
undefined |
| 3.1.12.19a2.24 |
$x^{12} + 3 x^{11} + 3 x^{8} + 24$ |
$S_3^2:C_6$ (as 12T121) |
$216$ |
$3$ |
$[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$ |
$[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$ |
$[\frac{5}{4},\frac{5}{4}]^{2}$ |
$[\frac{1}{4},\frac{1}{4}]^{2}$ |
$[8, 0]$ |
$[2, 1]$ |
$z^9 + z^6 + 1,z^2 + 2$ |
undefined |
