Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
3.6.6.1 |
$6$ |
$x^{6} - 6 x^{5} + 24 x^{4} + 6 x^{3} + 18 x + 9$ |
$3$ |
$3$ |
$2$ |
$6$ |
$C_3^2:C_4$ (as 6T10) |
$2$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 3 t x^{2} + \left(3 t + 6\right) x + 3$ |
$[1, 0]$ |
$[1]$ |
3.6.6.2 |
$6$ |
$x^{6} - 6 x^{5} + 39 x^{4} + 60 x^{3} - 18 x + 9$ |
$3$ |
$3$ |
$2$ |
$6$ |
$C_3^2:C_4$ (as 6T10) |
$2$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + \left(6 t + 3\right) x^{2} + 3 t x + 3$ |
$[1, 0]$ |
$[1]$ |
3.6.6.3 |
$6$ |
$x^{6} + 18 x^{5} + 120 x^{4} + 386 x^{3} + 723 x^{2} + 732 x + 305$ |
$3$ |
$3$ |
$2$ |
$6$ |
$D_{6}$ (as 6T3) |
$2$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 6 x^{2} + 3 x + 3$ |
$[1, 0]$ |
$[1]$ |
3.6.6.4 |
$6$ |
$x^{6} + 48 x^{4} + 6 x^{3} + 36 x^{2} + 36 x + 9$ |
$3$ |
$3$ |
$2$ |
$6$ |
$D_{6}$ (as 6T3) |
$2$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + \left(6 t + 6\right) x^{2} + 6 x + 3$ |
$[1, 0]$ |
$[1]$ |
3.6.6.5 |
$6$ |
$x^{6} + 6 x^{3} + 36 x^{2} + 9$ |
$3$ |
$3$ |
$2$ |
$6$ |
$S_3^2$ (as 6T9) |
$2$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + \left(6 t + 6\right) x + 3$ |
$[1, 0]$ |
$[1]$ |
3.6.8.1 |
$6$ |
$x^{6} - 6 x^{5} + 45 x^{4} + 6 x^{3} - 18 x^{2} + 9$ |
$3$ |
$3$ |
$2$ |
$8$ |
$C_3^2:C_4$ (as 6T10) |
$4$ |
$1$ |
$[2]$ |
$[2, 2]^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + \left(6 t + 3\right) x^{2} + 3$ |
$[2, 0]$ |
$[2]$ |
3.6.8.10 |
$6$ |
$x^{6} + 9 x^{4} + 42 x^{3} + 441$ |
$3$ |
$3$ |
$2$ |
$8$ |
$S_3\times C_3$ (as 6T5) |
$2$ |
$1$ |
$[2]$ |
$[2, 2]^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + \left(3 t + 3\right) x^{2} + 21$ |
$[2, 0]$ |
$[1]$ |
3.6.8.2 |
$6$ |
$x^{6} - 12 x^{5} + 72 x^{4} + 6 x^{3} + 72 x^{2} + 90$ |
$3$ |
$3$ |
$2$ |
$8$ |
$C_3^2:C_4$ (as 6T10) |
$4$ |
$1$ |
$[2]$ |
$[2, 2]^{4}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 6 t x^{2} + 9 t + 12$ |
$[2, 0]$ |
$[2]$ |
3.6.8.3 |
$6$ |
$x^{6} + 18 x^{5} + 114 x^{4} + 326 x^{3} + 570 x^{2} + 528 x + 197$ |
$3$ |
$3$ |
$2$ |
$8$ |
$C_6$ (as 6T1) |
$2$ |
$1$ |
$[2]$ |
$[2]^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 6 x^{2} + 3$ |
$[2, 0]$ |
$[1]$ |
3.6.8.4 |
$6$ |
$x^{6} + 18 x^{5} + 114 x^{4} + 344 x^{3} + 732 x^{2} + 744 x + 296$ |
$3$ |
$3$ |
$2$ |
$8$ |
$C_6$ (as 6T1) |
$2$ |
$1$ |
$[2]$ |
$[2]^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 6 x^{2} + 12$ |
$[2, 0]$ |
$[1]$ |
3.6.8.5 |
$6$ |
$x^{6} + 12 x^{5} + 57 x^{4} + 146 x^{3} + 240 x^{2} + 204 x + 65$ |
$3$ |
$3$ |
$2$ |
$8$ |
$S_3$ (as 6T2) |
$2$ |
$1$ |
$[2]$ |
$[2]^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 3 x^{2} + 3$ |
$[2, 0]$ |
$[1]$ |
3.6.8.6 |
$6$ |
$x^{6} + 18 x^{5} + 114 x^{4} + 362 x^{3} + 894 x^{2} + 960 x + 557$ |
$3$ |
$3$ |
$2$ |
$8$ |
$C_6$ (as 6T1) |
$2$ |
$1$ |
$[2]$ |
$[2]^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 6 x^{2} + 21$ |
$[2, 0]$ |
$[1]$ |
3.6.8.7 |
$6$ |
$x^{6} + 6 x^{5} + 9 x^{4} + 6 x^{3} + 18 x^{2} + 90$ |
$3$ |
$3$ |
$2$ |
$8$ |
$S_3\times C_3$ (as 6T5) |
$6$ |
$1$ |
$[2]$ |
$[2]^{6}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 3 x^{2} + 9 t + 12$ |
$[2, 0]$ |
$[1]$ |
3.6.8.8 |
$6$ |
$x^{6} + 9 x^{4} + 24 x^{3} + 144$ |
$3$ |
$3$ |
$2$ |
$8$ |
$S_3\times C_3$ (as 6T5) |
$2$ |
$1$ |
$[2]$ |
$[2, 2]^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + \left(3 t + 3\right) x^{2} + 12$ |
$[2, 0]$ |
$[1]$ |
3.6.8.9 |
$6$ |
$x^{6} + 36 x^{4} + 6 x^{3} + 9$ |
$3$ |
$3$ |
$2$ |
$8$ |
$S_3\times C_3$ (as 6T5) |
$2$ |
$1$ |
$[2]$ |
$[2, 2]^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + \left(6 t + 6\right) x^{2} + 3$ |
$[2, 0]$ |
$[1]$ |
3.6.10.1 |
$6$ |
$x^{6} + 18 x^{4} + 6 x^{3} + 162 x^{2} + 216 x + 90$ |
$3$ |
$3$ |
$2$ |
$10$ |
$D_{6}$ (as 6T3) |
$2$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + \left(9 t + 18\right) x + 9 t + 12$ |
$[3, 0]$ |
$[1]$ |
3.6.10.2 |
$6$ |
$x^{6} - 36 x^{4} - 12 x^{3} + 648 x^{2} + 864 x + 360$ |
$3$ |
$3$ |
$2$ |
$10$ |
$D_{6}$ (as 6T3) |
$2$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 18 t x + 18 t + 12$ |
$[3, 0]$ |
$[1]$ |
3.6.10.3 |
$6$ |
$x^{6} + 6 x^{5} + 36 x^{4} + 128 x^{3} + 297 x^{2} + 474 x + 482$ |
$3$ |
$3$ |
$2$ |
$10$ |
$D_{6}$ (as 6T3) |
$2$ |
$2$ |
$[5/2]$ |
$[5/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 9 x + 12$ |
$[3, 0]$ |
$[1]$ |
3.6.10.4 |
$6$ |
$x^{6} + 6 x^{3} + 81 x^{2} + 9$ |
$3$ |
$3$ |
$2$ |
$10$ |
$S_3^2$ (as 6T9) |
$2$ |
$2$ |
$[5/2]$ |
$[3/2, 5/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + \left(9 t + 9\right) x + 3$ |
$[3, 0]$ |
$[1]$ |
3.6.10.5 |
$6$ |
$x^{6} - 12 x^{3} + 360$ |
$3$ |
$3$ |
$2$ |
$10$ |
$S_3^2$ (as 6T9) |
$2$ |
$2$ |
$[5/2]$ |
$[3/2, 5/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 18 t + 12$ |
$[3, 0]$ |
$[1]$ |
3.6.10.6 |
$6$ |
$x^{6} - 18 x^{4} - 12 x^{3} + 162 x^{2} + 432 x + 360$ |
$3$ |
$3$ |
$2$ |
$10$ |
$S_3^2$ (as 6T9) |
$2$ |
$2$ |
$[5/2]$ |
$[3/2, 5/2]_{2}^{2}$ |
$t^{2} + 2 t + 2$ |
$x^{3} + 9 t x + 18 t + 12$ |
$[3, 0]$ |
$[1]$ |