Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
19.12.0.1 |
$12$ |
$x^{12} + 3 x^{7} + 2 x^{6} + 18 x^{5} + 2 x^{4} + 9 x^{3} + 16 x^{2} + 7 x + 2$ |
$19$ |
$1$ |
$12$ |
$0$ |
$C_{12}$ (as 12T1) |
$12$ |
$1$ |
$[\ ]$ |
$[\ ]^{12}$ |
$t^{12} + 3 t^{7} + 2 t^{6} + 18 t^{5} + 2 t^{4} + 9 t^{3} + 16 t^{2} + 7 t + 2$ |
$x - 19$ |
$[0]$ |
$[]$ |
19.12.6.1 |
$12$ |
$x^{12} + 114 x^{10} + 34 x^{9} + 5449 x^{8} + 12 x^{7} + 134889 x^{6} - 75118 x^{5} + 1847901 x^{4} - 1865072 x^{3} + 14269503 x^{2} - 12672520 x + 53461691$ |
$19$ |
$2$ |
$6$ |
$6$ |
$C_6\times C_2$ (as 12T2) |
$6$ |
$2$ |
$[\ ]$ |
$[\ ]_{2}^{6}$ |
$t^{6} + 17 t^{3} + 17 t^{2} + 6 t + 2$ |
$x^{2} + 19$ |
$[0]$ |
$[1]$ |
19.12.6.2 |
$12$ |
$x^{12} - 116603 x^{6} + 2215457 x^{4} - 14856594 x^{2} + 94091762$ |
$19$ |
$2$ |
$6$ |
$6$ |
$C_{12}$ (as 12T1) |
$6$ |
$2$ |
$[\ ]$ |
$[\ ]_{2}^{6}$ |
$t^{6} + 17 t^{3} + 17 t^{2} + 6 t + 2$ |
$x^{2} + 19 t$ |
$[0]$ |
$[1]$ |
19.12.8.1 |
$12$ |
$x^{12} + 386 x^{10} + 109 x^{9} + 55308 x^{8} + 21792 x^{7} + 3500499 x^{6} + 2034936 x^{5} + 84821873 x^{4} + 99877907 x^{3} + 174885148 x^{2} + 920938017 x + 335157671$ |
$19$ |
$3$ |
$4$ |
$8$ |
$C_{12}$ (as 12T1) |
$4$ |
$3$ |
$[\ ]$ |
$[\ ]_{3}^{4}$ |
$t^{4} + 2 t^{2} + 11 t + 2$ |
$x^{3} + 95 x + 19$ |
$[0]$ |
$[1]$ |
19.12.8.2 |
$12$ |
$x^{12} - 76 x^{9} + 2888 x^{6} + 775067 x^{3} + 521284$ |
$19$ |
$3$ |
$4$ |
$8$ |
$C_{12}$ (as 12T1) |
$4$ |
$3$ |
$[\ ]$ |
$[\ ]_{3}^{4}$ |
$t^{4} + 2 t^{2} + 11 t + 2$ |
$x^{3} + 19 t^{2}$ |
$[0]$ |
$[1]$ |
19.12.8.3 |
$12$ |
$x^{12} + 722 x^{6} - 75449 x^{3} + 260642$ |
$19$ |
$3$ |
$4$ |
$8$ |
$C_{12}$ (as 12T1) |
$4$ |
$3$ |
$[\ ]$ |
$[\ ]_{3}^{4}$ |
$t^{4} + 2 t^{2} + 11 t + 2$ |
$x^{3} + 19 t$ |
$[0]$ |
$[1]$ |
19.12.9.1 |
$12$ |
$x^{12} + 16 x^{10} + 68 x^{9} + 153 x^{8} + 816 x^{7} + 1382 x^{6} - 12240 x^{5} + 17643 x^{4} + 18836 x^{3} + 180010 x^{2} + 131580 x + 194652$ |
$19$ |
$4$ |
$3$ |
$9$ |
$D_4 \times C_3$ (as 12T14) |
$6$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{6}$ |
$t^{3} + 4 t + 17$ |
$x^{4} + 19$ |
$[0]$ |
$[2]$ |
19.12.9.2 |
$12$ |
$x^{12} + 1444 x^{4} - 116603$ |
$19$ |
$4$ |
$3$ |
$9$ |
$D_4 \times C_3$ (as 12T14) |
$6$ |
$4$ |
$[\ ]$ |
$[\ ]_{4}^{6}$ |
$t^{3} + 4 t + 17$ |
$x^{4} + 19 t$ |
$[0]$ |
$[2]$ |
19.12.10.1 |
$12$ |
$x^{12} + 108 x^{11} + 4872 x^{10} + 117720 x^{9} + 1613580 x^{8} + 12041568 x^{7} + 40427622 x^{6} + 24085188 x^{5} + 6545520 x^{4} + 3116880 x^{3} + 29259672 x^{2} + 208806768 x + 622519465$ |
$19$ |
$6$ |
$2$ |
$10$ |
$C_6\times C_2$ (as 12T2) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 18 t + 2$ |
$x^{6} + 19$ |
$[0]$ |
$[1]$ |
19.12.10.2 |
$12$ |
$x^{12} + 304 x^{6} - 5415$ |
$19$ |
$6$ |
$2$ |
$10$ |
$C_6\times C_2$ (as 12T2) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 18 t + 2$ |
$x^{6} + 19 t + 323$ |
$[0]$ |
$[1]$ |
19.12.10.3 |
$12$ |
$x^{12} - 5396 x^{6} - 21660$ |
$19$ |
$6$ |
$2$ |
$10$ |
$C_6\times C_2$ (as 12T2) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 18 t + 2$ |
$x^{6} + 304 t + 38$ |
$[0]$ |
$[1]$ |
19.12.10.4 |
$12$ |
$x^{12} - 342 x^{6} + 722$ |
$19$ |
$6$ |
$2$ |
$10$ |
$C_{12}$ (as 12T1) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 18 t + 2$ |
$x^{6} + 19 t$ |
$[0]$ |
$[1]$ |
19.12.10.5 |
$12$ |
$x^{12} - 5510 x^{6} - 1650131$ |
$19$ |
$6$ |
$2$ |
$10$ |
$C_{12}$ (as 12T1) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 18 t + 2$ |
$x^{6} + 342 t + 323$ |
$[0]$ |
$[1]$ |
19.12.10.6 |
$12$ |
$x^{12} - 5928 x^{6} - 454860$ |
$19$ |
$6$ |
$2$ |
$10$ |
$C_{12}$ (as 12T1) |
$2$ |
$6$ |
$[\ ]$ |
$[\ ]_{6}^{2}$ |
$t^{2} + 18 t + 2$ |
$x^{6} + 342 t + 114$ |
$[0]$ |
$[1]$ |
19.12.11.1 |
$12$ |
$x^{12} + 76$ |
$19$ |
$12$ |
$1$ |
$11$ |
$D_4 \times C_3$ (as 12T14) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 17$ |
$x^{12} + 76$ |
$[0]$ |
$[2]$ |
19.12.11.2 |
$12$ |
$x^{12} + 19$ |
$19$ |
$12$ |
$1$ |
$11$ |
$D_4 \times C_3$ (as 12T14) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 17$ |
$x^{12} + 19$ |
$[0]$ |
$[2]$ |
19.12.11.3 |
$12$ |
$x^{12} + 95$ |
$19$ |
$12$ |
$1$ |
$11$ |
$D_4 \times C_3$ (as 12T14) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 17$ |
$x^{12} + 95$ |
$[0]$ |
$[2]$ |
19.12.11.4 |
$12$ |
$x^{12} + 152$ |
$19$ |
$12$ |
$1$ |
$11$ |
$D_4 \times C_3$ (as 12T14) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 17$ |
$x^{12} + 152$ |
$[0]$ |
$[2]$ |
19.12.11.5 |
$12$ |
$x^{12} + 38$ |
$19$ |
$12$ |
$1$ |
$11$ |
$D_4 \times C_3$ (as 12T14) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 17$ |
$x^{12} + 38$ |
$[0]$ |
$[2]$ |
19.12.11.6 |
$12$ |
$x^{12} + 190$ |
$19$ |
$12$ |
$1$ |
$11$ |
$D_4 \times C_3$ (as 12T14) |
$2$ |
$12$ |
$[\ ]$ |
$[\ ]_{12}^{2}$ |
$t + 17$ |
$x^{12} + 190$ |
$[0]$ |
$[2]$ |