| 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 |
| 5.2.5.12a6.1 |
$10$ |
$( x^{2} + 4 x + 2 )^{5} + \left(10 x + 5\right) ( x^{2} + 4 x + 2 )^{2} + 5$ |
$5$ |
$2$ |
$5$ |
$12$ |
$C_5^2 : C_4$ (as 10T10) |
$2$ |
$2$ |
$[\frac{3}{2}]$ |
$[\frac{1}{2}]$ |
$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$ |
$[\frac{1}{2},\frac{1}{2}]_{2}^{2}$ |
$[\frac{3}{2}]_{2}$ |
$[\frac{1}{2}]_{2}$ |
$1$ |
$t^{2} + 4 t + 2$ |
$x^{5} + \left(10 t + 5\right) x^{2} + 5$ |
$[2, 0]$ |
$[1]$ |
$z^2 + (2 t + 2)$ |
undefined |
| 5.2.5.12a13.1 |
$10$ |
$( x^{2} + 4 x + 2 )^{5} + \left(15 x + 20\right) ( x^{2} + 4 x + 2 )^{2} + 5$ |
$5$ |
$2$ |
$5$ |
$12$ |
$C_5^2 : C_4$ (as 10T10) |
$2$ |
$2$ |
$[\frac{3}{2}]$ |
$[\frac{1}{2}]$ |
$[\frac{3}{2}, \frac{3}{2}]_{2}^{2}$ |
$[\frac{1}{2},\frac{1}{2}]_{2}^{2}$ |
$[\frac{3}{2}]_{2}$ |
$[\frac{1}{2}]_{2}$ |
$1$ |
$t^{2} + 4 t + 2$ |
$x^{5} + \left(10 t + 10\right) x^{2} + 5$ |
$[2, 0]$ |
$[1]$ |
$z^2 + (3 t + 3)$ |
undefined |
| 5.2.5.16a6.1 |
$10$ |
$( x^{2} + 4 x + 2 )^{5} + \left(5 x + 15\right) ( x^{2} + 4 x + 2 )^{4} + 5$ |
$5$ |
$2$ |
$5$ |
$16$ |
$C_5^2 : C_4$ (as 10T10) |
$4$ |
$1$ |
$[2]$ |
$[1]$ |
$[2, 2]^{4}$ |
$[1,1]^{4}$ |
$[2]^{2}$ |
$[1]^{2}$ |
$1$ |
$t^{2} + 4 t + 2$ |
$x^{5} + \left(5 t + 15\right) x^{4} + 5$ |
$[4, 0]$ |
$[2]$ |
$z^4 + (t + 1)$ |
undefined |
| 5.2.5.16a13.1 |
$10$ |
$( x^{2} + 4 x + 2 )^{5} + \left(20 x + 10\right) ( x^{2} + 4 x + 2 )^{4} + 5$ |
$5$ |
$2$ |
$5$ |
$16$ |
$C_5^2 : C_4$ (as 10T10) |
$4$ |
$1$ |
$[2]$ |
$[1]$ |
$[2, 2]^{4}$ |
$[1,1]^{4}$ |
$[2]^{2}$ |
$[1]^{2}$ |
$1$ |
$t^{2} + 4 t + 2$ |
$x^{5} + \left(5 t + 5\right) x^{4} + 5$ |
$[4, 0]$ |
$[2]$ |
$z^4 + (4 t + 4)$ |
undefined |
| 5.1.10.15a2.2 |
$10$ |
$x^{10} + 5 x^{7} + 10 x^{6} + 5$ |
$5$ |
$1$ |
$10$ |
$15$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{7}{4}]$ |
$[\frac{3}{4}]$ |
$[\frac{5}{4}, \frac{7}{4}]_{4}$ |
$[\frac{1}{4},\frac{3}{4}]_{4}$ |
$[\frac{5}{4}]_{2}$ |
$[\frac{1}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 5 x^{7} + 10 x^{6} + 5$ |
$[6, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
| 5.1.10.15a2.3 |
$10$ |
$x^{10} + 10 x^{7} + 10 x^{6} + 5$ |
$5$ |
$1$ |
$10$ |
$15$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{7}{4}]$ |
$[\frac{3}{4}]$ |
$[\frac{5}{4}, \frac{7}{4}]_{4}$ |
$[\frac{1}{4},\frac{3}{4}]_{4}$ |
$[\frac{5}{4}]_{2}$ |
$[\frac{1}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 10 x^{7} + 10 x^{6} + 5$ |
$[6, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
| 5.1.10.15a2.5 |
$10$ |
$x^{10} + 5 x^{7} + 15 x^{6} + 5$ |
$5$ |
$1$ |
$10$ |
$15$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{7}{4}]$ |
$[\frac{3}{4}]$ |
$[\frac{5}{4}, \frac{7}{4}]_{4}$ |
$[\frac{1}{4},\frac{3}{4}]_{4}$ |
$[\frac{5}{4}]_{2}$ |
$[\frac{1}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 5 x^{7} + 15 x^{6} + 5$ |
$[6, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 2$ |
undefined |
| 5.1.10.15a2.6 |
$10$ |
$x^{10} + 10 x^{7} + 15 x^{6} + 5$ |
$5$ |
$1$ |
$10$ |
$15$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{7}{4}]$ |
$[\frac{3}{4}]$ |
$[\frac{5}{4}, \frac{7}{4}]_{4}$ |
$[\frac{1}{4},\frac{3}{4}]_{4}$ |
$[\frac{5}{4}]_{2}$ |
$[\frac{1}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 10 x^{7} + 15 x^{6} + 5$ |
$[6, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 2$ |
undefined |
| 5.1.10.15a2.8 |
$10$ |
$x^{10} + 5 x^{7} + 5 x^{6} + 10$ |
$5$ |
$1$ |
$10$ |
$15$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{7}{4}]$ |
$[\frac{3}{4}]$ |
$[\frac{5}{4}, \frac{7}{4}]_{4}$ |
$[\frac{1}{4},\frac{3}{4}]_{4}$ |
$[\frac{5}{4}]_{2}$ |
$[\frac{1}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 5 x^{7} + 5 x^{6} + 10$ |
$[6, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 2$ |
undefined |
| 5.1.10.15a2.9 |
$10$ |
$x^{10} + 10 x^{7} + 5 x^{6} + 10$ |
$5$ |
$1$ |
$10$ |
$15$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{7}{4}]$ |
$[\frac{3}{4}]$ |
$[\frac{5}{4}, \frac{7}{4}]_{4}$ |
$[\frac{1}{4},\frac{3}{4}]_{4}$ |
$[\frac{5}{4}]_{2}$ |
$[\frac{1}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 10 x^{7} + 5 x^{6} + 10$ |
$[6, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 2$ |
undefined |
| 5.1.10.15a2.11 |
$10$ |
$x^{10} + 5 x^{7} + 20 x^{6} + 10$ |
$5$ |
$1$ |
$10$ |
$15$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{7}{4}]$ |
$[\frac{3}{4}]$ |
$[\frac{5}{4}, \frac{7}{4}]_{4}$ |
$[\frac{1}{4},\frac{3}{4}]_{4}$ |
$[\frac{5}{4}]_{2}$ |
$[\frac{1}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 5 x^{7} + 20 x^{6} + 10$ |
$[6, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
| 5.1.10.15a2.12 |
$10$ |
$x^{10} + 10 x^{7} + 20 x^{6} + 10$ |
$5$ |
$1$ |
$10$ |
$15$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{7}{4}]$ |
$[\frac{3}{4}]$ |
$[\frac{5}{4}, \frac{7}{4}]_{4}$ |
$[\frac{1}{4},\frac{3}{4}]_{4}$ |
$[\frac{5}{4}]_{2}$ |
$[\frac{1}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 10 x^{7} + 20 x^{6} + 10$ |
$[6, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
| 5.1.10.19a2.6 |
$10$ |
$x^{10} + 25 x + 5$ |
$5$ |
$1$ |
$10$ |
$19$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{7}{4}, \frac{9}{4}]_{4}$ |
$[\frac{3}{4},\frac{5}{4}]_{4}$ |
$[\frac{7}{4}]_{2}$ |
$[\frac{3}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 25 x + 5$ |
$[10, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
| 5.1.10.19a2.7 |
$10$ |
$x^{10} + 25 x^{2} + 25 x + 5$ |
$5$ |
$1$ |
$10$ |
$19$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{7}{4}, \frac{9}{4}]_{4}$ |
$[\frac{3}{4},\frac{5}{4}]_{4}$ |
$[\frac{7}{4}]_{2}$ |
$[\frac{3}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 25 x^{2} + 25 x + 5$ |
$[10, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
| 5.1.10.19a2.8 |
$10$ |
$x^{10} + 50 x^{2} + 25 x + 5$ |
$5$ |
$1$ |
$10$ |
$19$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{7}{4}, \frac{9}{4}]_{4}$ |
$[\frac{3}{4},\frac{5}{4}]_{4}$ |
$[\frac{7}{4}]_{2}$ |
$[\frac{3}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 50 x^{2} + 25 x + 5$ |
$[10, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
| 5.1.10.19a2.9 |
$10$ |
$x^{10} + 75 x^{2} + 25 x + 5$ |
$5$ |
$1$ |
$10$ |
$19$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{7}{4}, \frac{9}{4}]_{4}$ |
$[\frac{3}{4},\frac{5}{4}]_{4}$ |
$[\frac{7}{4}]_{2}$ |
$[\frac{3}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 75 x^{2} + 25 x + 5$ |
$[10, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
| 5.1.10.19a2.10 |
$10$ |
$x^{10} + 100 x^{2} + 25 x + 5$ |
$5$ |
$1$ |
$10$ |
$19$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{7}{4}, \frac{9}{4}]_{4}$ |
$[\frac{3}{4},\frac{5}{4}]_{4}$ |
$[\frac{7}{4}]_{2}$ |
$[\frac{3}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 100 x^{2} + 25 x + 5$ |
$[10, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
| 5.1.10.19a2.11 |
$10$ |
$x^{10} + 50 x + 5$ |
$5$ |
$1$ |
$10$ |
$19$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{7}{4}, \frac{9}{4}]_{4}$ |
$[\frac{3}{4},\frac{5}{4}]_{4}$ |
$[\frac{7}{4}]_{2}$ |
$[\frac{3}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 50 x + 5$ |
$[10, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
| 5.1.10.19a2.12 |
$10$ |
$x^{10} + 25 x^{2} + 50 x + 5$ |
$5$ |
$1$ |
$10$ |
$19$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{7}{4}, \frac{9}{4}]_{4}$ |
$[\frac{3}{4},\frac{5}{4}]_{4}$ |
$[\frac{7}{4}]_{2}$ |
$[\frac{3}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 25 x^{2} + 50 x + 5$ |
$[10, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
| 5.1.10.19a2.13 |
$10$ |
$x^{10} + 50 x^{2} + 50 x + 5$ |
$5$ |
$1$ |
$10$ |
$19$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{7}{4}, \frac{9}{4}]_{4}$ |
$[\frac{3}{4},\frac{5}{4}]_{4}$ |
$[\frac{7}{4}]_{2}$ |
$[\frac{3}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 50 x^{2} + 50 x + 5$ |
$[10, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
| 5.1.10.19a2.14 |
$10$ |
$x^{10} + 75 x^{2} + 50 x + 5$ |
$5$ |
$1$ |
$10$ |
$19$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{7}{4}, \frac{9}{4}]_{4}$ |
$[\frac{3}{4},\frac{5}{4}]_{4}$ |
$[\frac{7}{4}]_{2}$ |
$[\frac{3}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 75 x^{2} + 50 x + 5$ |
$[10, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
| 5.1.10.19a2.15 |
$10$ |
$x^{10} + 100 x^{2} + 50 x + 5$ |
$5$ |
$1$ |
$10$ |
$19$ |
$C_5^2 : C_4$ (as 10T10) |
$1$ |
$4$ |
$[\frac{9}{4}]$ |
$[\frac{5}{4}]$ |
$[\frac{7}{4}, \frac{9}{4}]_{4}$ |
$[\frac{3}{4},\frac{5}{4}]_{4}$ |
$[\frac{7}{4}]_{2}$ |
$[\frac{3}{4}]_{2}$ |
$1$ |
$t + 3$ |
$x^{10} + 100 x^{2} + 50 x + 5$ |
$[10, 0]$ |
$[1, 1]$ |
$z^5 + 2,2 z^2 + 3$ |
undefined |
