| 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 |
| 5.16.1.0a1.1 |
$16$ |
$x^{16} + x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 2 x^{4} + 4 x^{3} + 4 x^{2} + x + 2$ |
$5$ |
$16$ |
$1$ |
$0$ |
$C_{16}$ (as 16T1) |
$16$ |
$1$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]^{16}$ |
$[\ ]^{16}$ |
$[\ ]$ |
$[\ ]$ |
$16$ |
$t^{16} + t^{8} + 4 t^{7} + 4 t^{6} + 4 t^{5} + 2 t^{4} + 4 t^{3} + 4 t^{2} + t + 2$ |
$x - 5$ |
$[0]$ |
$[\ ]$ |
|
undefined |
| 5.8.2.8a1.1 |
$16$ |
$( x^{8} + x^{4} + 3 x^{2} + 4 x + 2 )^{2} + 5 x$ |
$5$ |
$8$ |
$2$ |
$8$ |
$C_{16}$ (as 16T1) |
$8$ |
$2$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{2}^{8}$ |
$[\ ]_{2}^{8}$ |
$[\ ]$ |
$[\ ]$ |
$16$ |
$t^{8} + t^{4} + 3 t^{2} + 4 t + 2$ |
$x^{2} + 5 t$ |
$[0]$ |
$[1]$ |
$z + 2$ |
undefined |
| 5.8.2.8a1.2 |
$16$ |
$( x^{8} + x^{4} + 3 x^{2} + 4 x + 2 )^{2} + 5$ |
$5$ |
$8$ |
$2$ |
$8$ |
$C_8\times C_2$ (as 16T5) |
$8$ |
$2$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{2}^{8}$ |
$[\ ]_{2}^{8}$ |
$[\ ]$ |
$[\ ]$ |
$16$ |
$t^{8} + t^{4} + 3 t^{2} + 4 t + 2$ |
$x^{2} + 5$ |
$[0]$ |
$[1]$ |
$z + 2$ |
undefined |
| 5.4.4.12a1.1 |
$16$ |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5 x^{3}$ |
$5$ |
$4$ |
$4$ |
$12$ |
$C_{16}$ (as 16T1) |
$4$ |
$4$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{4}^{4}$ |
$[\ ]_{4}^{4}$ |
$[\ ]$ |
$[\ ]$ |
$16$ |
$t^{4} + 4 t^{2} + 4 t + 2$ |
$x^{4} + 5 t^{3}$ |
$[0]$ |
$[1]$ |
$z^3 + 4 z^2 + z + 4$ |
undefined |
| 5.4.4.12a1.2 |
$16$ |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5 x^{2}$ |
$5$ |
$4$ |
$4$ |
$12$ |
$C_8\times C_2$ (as 16T5) |
$4$ |
$4$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{4}^{4}$ |
$[\ ]_{4}^{4}$ |
$[\ ]$ |
$[\ ]$ |
$16$ |
$t^{4} + 4 t^{2} + 4 t + 2$ |
$x^{4} + 5 t^{2}$ |
$[0]$ |
$[1]$ |
$z^3 + 4 z^2 + z + 4$ |
undefined |
| 5.4.4.12a1.3 |
$16$ |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5 x$ |
$5$ |
$4$ |
$4$ |
$12$ |
$C_{16}$ (as 16T1) |
$4$ |
$4$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{4}^{4}$ |
$[\ ]_{4}^{4}$ |
$[\ ]$ |
$[\ ]$ |
$16$ |
$t^{4} + 4 t^{2} + 4 t + 2$ |
$x^{4} + 5 t$ |
$[0]$ |
$[1]$ |
$z^3 + 4 z^2 + z + 4$ |
undefined |
| 5.4.4.12a1.4 |
$16$ |
$( x^{4} + 4 x^{2} + 4 x + 2 )^{4} + 5$ |
$5$ |
$4$ |
$4$ |
$12$ |
$C_4^2$ (as 16T4) |
$4$ |
$4$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{4}^{4}$ |
$[\ ]_{4}^{4}$ |
$[\ ]$ |
$[\ ]$ |
$16$ |
$t^{4} + 4 t^{2} + 4 t + 2$ |
$x^{4} + 5$ |
$[0]$ |
$[1]$ |
$z^3 + 4 z^2 + z + 4$ |
$[1]$ |
| 5.2.8.14a1.1 |
$16$ |
$( x^{2} + 4 x + 2 )^{8} + 5 x$ |
$5$ |
$2$ |
$8$ |
$14$ |
$C_{16} : C_2$ (as 16T22) |
$4$ |
$8$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{8}^{4}$ |
$[\ ]_{8}^{4}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$8$ |
$t^{2} + 4 t + 2$ |
$x^{8} + 5 t$ |
$[0]$ |
$[1]$ |
$z^7 + 3 z^6 + 3 z^5 + z^4 + z^2 + 3 z + 3$ |
undefined |
| 5.2.8.14a1.2 |
$16$ |
$( x^{2} + 4 x + 2 )^{8} + 10 x$ |
$5$ |
$2$ |
$8$ |
$14$ |
$C_{16} : C_2$ (as 16T22) |
$4$ |
$8$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{8}^{4}$ |
$[\ ]_{8}^{4}$ |
$[\ ]^{2}$ |
$[\ ]^{2}$ |
$8$ |
$t^{2} + 4 t + 2$ |
$x^{8} + 10 t$ |
$[0]$ |
$[1]$ |
$z^7 + 3 z^6 + 3 z^5 + z^4 + z^2 + 3 z + 3$ |
undefined |
| 5.2.8.14a1.3 |
$16$ |
$( x^{2} + 4 x + 2 )^{8} + 5$ |
$5$ |
$2$ |
$8$ |
$14$ |
$C_8: C_2$ (as 16T6) |
$2$ |
$8$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{8}^{2}$ |
$[\ ]_{8}^{2}$ |
$[\ ]$ |
$[\ ]$ |
$16$ |
$t^{2} + 4 t + 2$ |
$x^{8} + 5$ |
$[0]$ |
$[1]$ |
$z^7 + 3 z^6 + 3 z^5 + z^4 + z^2 + 3 z + 3$ |
$[2]$ |
| 5.2.8.14a1.4 |
$16$ |
$( x^{2} + 4 x + 2 )^{8} + 10$ |
$5$ |
$2$ |
$8$ |
$14$ |
$C_8: C_2$ (as 16T6) |
$2$ |
$8$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{8}^{2}$ |
$[\ ]_{8}^{2}$ |
$[\ ]$ |
$[\ ]$ |
$16$ |
$t^{2} + 4 t + 2$ |
$x^{8} + 10$ |
$[0]$ |
$[1]$ |
$z^7 + 3 z^6 + 3 z^5 + z^4 + z^2 + 3 z + 3$ |
undefined |
| 5.2.8.14a1.5 |
$16$ |
$( x^{2} + 4 x + 2 )^{8} + 10 x + 10$ |
$5$ |
$2$ |
$8$ |
$14$ |
$C_8: C_2$ (as 16T6) |
$2$ |
$8$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{8}^{2}$ |
$[\ ]_{8}^{2}$ |
$[\ ]$ |
$[\ ]$ |
$16$ |
$t^{2} + 4 t + 2$ |
$x^{8} + 10 t + 10$ |
$[0]$ |
$[1]$ |
$z^7 + 3 z^6 + 3 z^5 + z^4 + z^2 + 3 z + 3$ |
$[2]$ |
| 5.2.8.14a1.6 |
$16$ |
$( x^{2} + 4 x + 2 )^{8} + 5 x + 15$ |
$5$ |
$2$ |
$8$ |
$14$ |
$C_8: C_2$ (as 16T6) |
$2$ |
$8$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{8}^{2}$ |
$[\ ]_{8}^{2}$ |
$[\ ]$ |
$[\ ]$ |
$16$ |
$t^{2} + 4 t + 2$ |
$x^{8} + 5 t + 15$ |
$[0]$ |
$[1]$ |
$z^7 + 3 z^6 + 3 z^5 + z^4 + z^2 + 3 z + 3$ |
undefined |
| 5.1.16.15a1.1 |
$16$ |
$x^{16} + 5$ |
$5$ |
$1$ |
$16$ |
$15$ |
$C_{16}:C_4$ (as 16T125) |
$4$ |
$16$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{16}^{4}$ |
$[\ ]_{16}^{4}$ |
$[\ ]^{4}$ |
$[\ ]^{4}$ |
$4$ |
$t + 3$ |
$x^{16} + 5$ |
$[0]$ |
$[4]$ |
$z^{15} + z^{14} + 3 z^{10} + 3 z^9 + 3 z^5 + 3 z^4 + 1$ |
$[4]$ |
| 5.1.16.15a1.2 |
$16$ |
$x^{16} + 10$ |
$5$ |
$1$ |
$16$ |
$15$ |
$C_{16}:C_4$ (as 16T125) |
$4$ |
$16$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{16}^{4}$ |
$[\ ]_{16}^{4}$ |
$[\ ]^{4}$ |
$[\ ]^{4}$ |
$4$ |
$t + 3$ |
$x^{16} + 10$ |
$[0]$ |
$[4]$ |
$z^{15} + z^{14} + 3 z^{10} + 3 z^9 + 3 z^5 + 3 z^4 + 1$ |
undefined |
| 5.1.16.15a1.3 |
$16$ |
$x^{16} + 15$ |
$5$ |
$1$ |
$16$ |
$15$ |
$C_{16}:C_4$ (as 16T125) |
$4$ |
$16$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{16}^{4}$ |
$[\ ]_{16}^{4}$ |
$[\ ]^{4}$ |
$[\ ]^{4}$ |
$4$ |
$t + 3$ |
$x^{16} + 15$ |
$[0]$ |
$[4]$ |
$z^{15} + z^{14} + 3 z^{10} + 3 z^9 + 3 z^5 + 3 z^4 + 1$ |
undefined |
| 5.1.16.15a1.4 |
$16$ |
$x^{16} + 20$ |
$5$ |
$1$ |
$16$ |
$15$ |
$C_{16}:C_4$ (as 16T125) |
$4$ |
$16$ |
$[\ ]$ |
$[\ ]$ |
$[\ ]_{16}^{4}$ |
$[\ ]_{16}^{4}$ |
$[\ ]^{4}$ |
$[\ ]^{4}$ |
$4$ |
$t + 3$ |
$x^{16} + 20$ |
$[0]$ |
$[4]$ |
$z^{15} + z^{14} + 3 z^{10} + 3 z^9 + 3 z^5 + 3 z^4 + 1$ |
undefined |
