| Label |
$n$ |
Polynomial |
$p$ |
$f$ |
$e$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible Artin slopes |
Visible Swan slopes |
Artin slope content |
Swan slope content |
Hidden Artin slopes |
Hidden Swan slopes |
$\#\Aut(K/\Q_p)$ |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
Resid. Poly |
Jump Set |
| 5.12.1.0a1.1 |
$12$ |
$x^{12} + x^{7} + x^{6} + 4 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 2$ |
$5$ |
$12$ |
$1$ |
$0$ |
$C_{12}$ (as 12T1) |
$12$ |
$1$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]^{12}$ |
$[\ ]^{12}$ |
$[\ ]$ |
$[\ ]$ |
$12$ |
$t^{12} + t^{7} + t^{6} + 4 t^{4} + 4 t^{3} + 3 t^{2} + 2 t + 2$ |
$x - 5$ |
$[0]$ |
$[\ ]$ |
|
undefined |
| 5.6.2.6a1.1 |
$12$ |
$( x^{6} + x^{4} + 4 x^{3} + x^{2} + 2 )^{2} + 5 x$ |
$5$ |
$6$ |
$2$ |
$6$ |
$C_{12}$ (as 12T1) |
$6$ |
$2$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{2}^{6}$ |
$[\ ]_{2}^{6}$ |
$[\ ]$ |
$[\ ]$ |
$12$ |
$t^{6} + t^{4} + 4 t^{3} + t^{2} + 2$ |
$x^{2} + 5 t$ |
$[0]$ |
$[1]$ |
$z + 2$ |
undefined |
| 5.6.2.6a1.2 |
$12$ |
$( x^{6} + x^{4} + 4 x^{3} + x^{2} + 2 )^{2} + 5$ |
$5$ |
$6$ |
$2$ |
$6$ |
$C_6\times C_2$ (as 12T2) |
$6$ |
$2$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{2}^{6}$ |
$[\ ]_{2}^{6}$ |
$[\ ]$ |
$[\ ]$ |
$12$ |
$t^{6} + t^{4} + 4 t^{3} + t^{2} + 2$ |
$x^{2} + 5$ |
$[0]$ |
$[1]$ |
$z + 2$ |
undefined |
| 5.4.3.8a1.1 |
$12$ |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{3} + 5 x$ |
$5$ |
$4$ |
$3$ |
$8$ |
$C_3\times (C_3 : C_4)$ (as 12T19) |
$12$ |
$3$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{3}^{12}$ |
$[\ ]_{3}^{12}$ |
$[\ ]^{3}$ |
$[\ ]^{3}$ |
$6$ |
$t^{4} + 4 t^{2} + 4 t + 2$ |
$x^{3} + 5 t$ |
$[0]$ |
$[1]$ |
$z^2 + 3 z + 3$ |
undefined |
| 5.4.3.8a1.2 |
$12$ |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{3} + 5$ |
$5$ |
$4$ |
$3$ |
$8$ |
$C_3 : C_4$ (as 12T5) |
$4$ |
$3$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{3}^{4}$ |
$[\ ]_{3}^{4}$ |
$[\ ]$ |
$[\ ]$ |
$12$ |
$t^{4} + 4 t^{2} + 4 t + 2$ |
$x^{3} + 5$ |
$[0]$ |
$[1]$ |
$z^2 + 3 z + 3$ |
undefined |
| 5.3.4.9a1.1 |
$12$ |
$( x^{3} + 3 x + 3 )^{4} + 5 x^{2}$ |
$5$ |
$3$ |
$4$ |
$9$ |
$C_{12}$ (as 12T1) |
$3$ |
$4$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{4}^{3}$ |
$[\ ]_{4}^{3}$ |
$[\ ]$ |
$[\ ]$ |
$12$ |
$t^{3} + 3 t + 3$ |
$x^{4} + 5 t^{2}$ |
$[0]$ |
$[1]$ |
$z^3 + 4 z^2 + z + 4$ |
undefined |
| 5.3.4.9a1.2 |
$12$ |
$( x^{3} + 3 x + 3 )^{4} + 5 x$ |
$5$ |
$3$ |
$4$ |
$9$ |
$C_{12}$ (as 12T1) |
$3$ |
$4$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{4}^{3}$ |
$[\ ]_{4}^{3}$ |
$[\ ]$ |
$[\ ]$ |
$12$ |
$t^{3} + 3 t + 3$ |
$x^{4} + 5 t$ |
$[0]$ |
$[1]$ |
$z^3 + 4 z^2 + z + 4$ |
undefined |
| 5.3.4.9a1.3 |
$12$ |
$( x^{3} + 3 x + 3 )^{4} + 5$ |
$5$ |
$3$ |
$4$ |
$9$ |
$C_{12}$ (as 12T1) |
$3$ |
$4$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{4}^{3}$ |
$[\ ]_{4}^{3}$ |
$[\ ]$ |
$[\ ]$ |
$12$ |
$t^{3} + 3 t + 3$ |
$x^{4} + 5$ |
$[0]$ |
$[1]$ |
$z^3 + 4 z^2 + z + 4$ |
$[1]$ |
| 5.3.4.9a1.4 |
$12$ |
$( x^{3} + 3 x + 3 )^{4} + 10 x + 10$ |
$5$ |
$3$ |
$4$ |
$9$ |
$C_{12}$ (as 12T1) |
$3$ |
$4$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{4}^{3}$ |
$[\ ]_{4}^{3}$ |
$[\ ]$ |
$[\ ]$ |
$12$ |
$t^{3} + 3 t + 3$ |
$x^{4} + 10 t + 10$ |
$[0]$ |
$[1]$ |
$z^3 + 4 z^2 + z + 4$ |
undefined |
| 5.2.6.10a1.1 |
$12$ |
$( x^{2} + 4 x + 2 )^{6} + 5 x$ |
$5$ |
$2$ |
$6$ |
$10$ |
$C_3\times (C_3 : C_4)$ (as 12T19) |
$6$ |
$6$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{6}^{6}$ |
$[\ ]_{6}^{6}$ |
$[\ ]^{3}$ |
$[\ ]^{3}$ |
$6$ |
$t^{2} + 4 t + 2$ |
$x^{6} + 20 t + 5$ |
$[0]$ |
$[1]$ |
$z^5 + z^4 + 1$ |
undefined |
| 5.2.6.10a1.2 |
$12$ |
$( x^{2} + 4 x + 2 )^{6} + 5$ |
$5$ |
$2$ |
$6$ |
$10$ |
$D_6$ (as 12T3) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$[\ ]_{6}^{2}$ |
$[\ ]$ |
$[\ ]$ |
$12$ |
$t^{2} + 4 t + 2$ |
$x^{6} + 5$ |
$[0]$ |
$[1]$ |
$z^5 + z^4 + 1$ |
undefined |
| 5.2.6.10a1.3 |
$12$ |
$( x^{2} + 4 x + 2 )^{6} + 5 x + 15$ |
$5$ |
$2$ |
$6$ |
$10$ |
$C_6\times S_3$ (as 12T18) |
$6$ |
$6$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{6}^{6}$ |
$[\ ]_{6}^{6}$ |
$[\ ]^{3}$ |
$[\ ]^{3}$ |
$6$ |
$t^{2} + 4 t + 2$ |
$x^{6} + 10 t + 10$ |
$[0]$ |
$[1]$ |
$z^5 + z^4 + 1$ |
undefined |
| 5.2.6.10a1.4 |
$12$ |
$( x^{2} + 4 x + 2 )^{6} + 20 x + 15$ |
$5$ |
$2$ |
$6$ |
$10$ |
$C_3 : C_4$ (as 12T5) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$[\ ]_{6}^{2}$ |
$[\ ]$ |
$[\ ]$ |
$12$ |
$t^{2} + 4 t + 2$ |
$x^{6} + 20 t + 15$ |
$[0]$ |
$[1]$ |
$z^5 + z^4 + 1$ |
undefined |
| 5.1.12.11a1.1 |
$12$ |
$x^{12} + 5$ |
$5$ |
$1$ |
$12$ |
$11$ |
$S_3 \times C_4$ (as 12T11) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$[\ ]_{12}^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$4$ |
$t + 3$ |
$x^{12} + 5$ |
$[0]$ |
$[2]$ |
$z^{11} + 2 z^{10} + z^9 + 2 z^6 + 4 z^5 + 2 z^4 + z + 2$ |
$[3]$ |
| 5.1.12.11a1.2 |
$12$ |
$x^{12} + 10$ |
$5$ |
$1$ |
$12$ |
$11$ |
$S_3 \times C_4$ (as 12T11) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$[\ ]_{12}^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$4$ |
$t + 3$ |
$x^{12} + 10$ |
$[0]$ |
$[2]$ |
$z^{11} + 2 z^{10} + z^9 + 2 z^6 + 4 z^5 + 2 z^4 + z + 2$ |
undefined |
| 5.1.12.11a1.3 |
$12$ |
$x^{12} + 15$ |
$5$ |
$1$ |
$12$ |
$11$ |
$S_3 \times C_4$ (as 12T11) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$[\ ]_{12}^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$4$ |
$t + 3$ |
$x^{12} + 15$ |
$[0]$ |
$[2]$ |
$z^{11} + 2 z^{10} + z^9 + 2 z^6 + 4 z^5 + 2 z^4 + z + 2$ |
undefined |
| 5.1.12.11a1.4 |
$12$ |
$x^{12} + 20$ |
$5$ |
$1$ |
$12$ |
$11$ |
$S_3 \times C_4$ (as 12T11) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$[\ ]_{12}^{2}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$4$ |
$t + 3$ |
$x^{12} + 20$ |
$[0]$ |
$[2]$ |
$z^{11} + 2 z^{10} + z^9 + 2 z^6 + 4 z^5 + 2 z^4 + z + 2$ |
undefined |
