| Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
| 19.8.0.1 |
$8$ |
$x^{8} + x^{4} + 12 x^{3} + 10 x^{2} + 3 x + 2$ |
$19$ |
$1$ |
$8$ |
$0$ |
$C_8$ (as 8T1) |
$8$ |
$1$ |
$[\ ]$ |
$[\ ]^{8}$ |
$t^{8} + t^{4} + 12 t^{3} + 10 t^{2} + 3 t + 2$ |
$x - 19$ |
$[0]$ |
$[]$ |
| 19.8.4.1 |
$8$ |
$x^{8} + 80 x^{6} + 22 x^{5} + 2250 x^{4} - 792 x^{3} + 25817 x^{2} - 22946 x + 107924$ |
$19$ |
$2$ |
$4$ |
$4$ |
$C_4\times C_2$ (as 8T2) |
$4$ |
$2$ |
$[\ ]$ |
$[\ ]_{2}^{4}$ |
$t^{4} + 2 t^{2} + 11 t + 2$ |
$x^{2} + 19$ |
$[0]$ |
$[1]$ |
| 19.8.4.2 |
$8$ |
$x^{8} + 722 x^{4} - 75449 x^{2} + 260642$ |
$19$ |
$2$ |
$4$ |
$4$ |
$C_8$ (as 8T1) |
$4$ |
$2$ |
$[\ ]$ |
$[\ ]_{2}^{4}$ |
$t^{4} + 2 t^{2} + 11 t + 2$ |
$x^{2} + 19 t$ |
$[0]$ |
$[1]$ |
| 19.8.6.1 |
$8$ |
$x^{8} + 304 x^{4} - 5415$ |
$19$ |
$4$ |
$2$ |
$6$ |
$Q_8$ (as 8T5) |
$2$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{2}$ |
$t^{2} + 18 t + 2$ |
$x^{4} + 19 t + 323$ |
$[0]$ |
$[1]$ |
| 19.8.6.2 |
$8$ |
$x^{8} + 72 x^{7} + 1952 x^{6} + 23760 x^{5} + 112814 x^{4} + 48888 x^{3} + 44288 x^{2} + 435600 x + 1945825$ |
$19$ |
$4$ |
$2$ |
$6$ |
$D_4$ (as 8T4) |
$2$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{2}$ |
$t^{2} + 18 t + 2$ |
$x^{4} + 19$ |
$[0]$ |
$[1]$ |
| 19.8.6.3 |
$8$ |
$x^{8} - 5510 x^{4} - 1650131$ |
$19$ |
$4$ |
$2$ |
$6$ |
$C_8:C_2$ (as 8T7) |
$4$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{4}$ |
$t^{2} + 18 t + 2$ |
$x^{4} + 342 t + 323$ |
$[0]$ |
$[1]$ |
| 19.8.7.1 |
$8$ |
$x^{8} + 19$ |
$19$ |
$8$ |
$1$ |
$7$ |
$QD_{16}$ (as 8T8) |
$2$ |
$8$ |
$[\ ]$ |
$[\ ]_{8}^{2}$ |
$t + 17$ |
$x^{8} + 19$ |
$[0]$ |
$[2]$ |
| 19.8.7.2 |
$8$ |
$x^{8} + 38$ |
$19$ |
$8$ |
$1$ |
$7$ |
$QD_{16}$ (as 8T8) |
$2$ |
$8$ |
$[\ ]$ |
$[\ ]_{8}^{2}$ |
$t + 17$ |
$x^{8} + 38$ |
$[0]$ |
$[2]$ |
