| Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Unram. Ext. |
Eisen. Poly. |
Ind. of Insep. |
Assoc. Inertia |
| 3.9.9.1 |
$9$ |
$x^{9} + 90 x^{7} - 207 x^{6} + 540 x^{5} + 324 x^{4} + 243 x^{3} + 324 x^{2} + 162 x + 27$ |
$3$ |
$3$ |
$3$ |
$9$ |
$S_3\times C_3$ (as 9T4) |
$3$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{3}$ |
$t^{3} + 2 t + 1$ |
$x^{3} + 6 t x^{2} + 6 x + 3$ |
$[1, 0]$ |
$[1]$ |
| 3.9.9.10 |
$9$ |
$x^{9} - 24 x^{8} + 321 x^{7} - 117 x^{6} - 243 x^{5} + 1062 x^{4} + 1053 x^{3} + 135 x^{2} - 27 x + 27$ |
$3$ |
$3$ |
$3$ |
$9$ |
$(C_3^3:C_3):C_2$ (as 9T22) |
$3$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2, 3/2]_{2}^{3}$ |
$t^{3} + 2 t + 1$ |
$x^{3} + \left(6 t^{2} + 6 t\right) x^{2} + \left(3 t^{2} + 6 t + 3\right) x + 3$ |
$[1, 0]$ |
$[1]$ |
| 3.9.9.11 |
$9$ |
$x^{9} + 3 x + 3$ |
$3$ |
$9$ |
$1$ |
$9$ |
$(C_3^2:C_8):C_2$ (as 9T19) |
$2$ |
$8$ |
$[9/8, 9/8]$ |
$[9/8, 9/8]_{8}^{2}$ |
$t + 1$ |
$x^{9} + 3 x + 3$ |
$[1, 1, 0]$ |
$[1]$ |
| 3.9.9.12 |
$9$ |
$x^{9} + 6 x + 3$ |
$3$ |
$9$ |
$1$ |
$9$ |
$(C_3^2:C_8):C_2$ (as 9T19) |
$2$ |
$8$ |
$[9/8, 9/8]$ |
$[9/8, 9/8]_{8}^{2}$ |
$t + 1$ |
$x^{9} + 6 x + 3$ |
$[1, 1, 0]$ |
$[1]$ |
| 3.9.9.2 |
$9$ |
$x^{9} - 3 x^{7} + 9 x^{6} - 9 x^{5} - 18 x^{4} + 81 x^{3} - 27 x^{2} - 27 x + 27$ |
$3$ |
$3$ |
$3$ |
$9$ |
$C_3^2 : S_3 $ (as 9T13) |
$3$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{3}$ |
$t^{3} + 2 t + 1$ |
$x^{3} + \left(3 t^{2} + 3\right) x + 3$ |
$[1, 0]$ |
$[1]$ |
| 3.9.9.3 |
$9$ |
$x^{9} - 3 x^{8} + 126 x^{7} + 198 x^{6} + 297 x^{5} + 567 x^{4} + 324 x^{3} + 108 x^{2} + 81 x + 27$ |
$3$ |
$3$ |
$3$ |
$9$ |
$(C_3^3:C_3):C_2$ (as 9T22) |
$3$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2, 3/2]_{2}^{3}$ |
$t^{3} + 2 t + 1$ |
$x^{3} + \left(3 t^{2} + 6 t + 3\right) x^{2} + \left(3 t + 3\right) x + 3$ |
$[1, 0]$ |
$[1]$ |
| 3.9.9.4 |
$9$ |
$x^{9} - 24 x^{8} + 318 x^{7} - 189 x^{6} + 1080 x^{5} + 2826 x^{4} + 1350 x^{3} - 108 x^{2} - 54 x + 27$ |
$3$ |
$3$ |
$3$ |
$9$ |
$(C_3^3:C_3):C_2$ (as 9T22) |
$3$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2, 3/2]_{2}^{3}$ |
$t^{3} + 2 t + 1$ |
$x^{3} + \left(6 t^{2} + 6 t\right) x^{2} + \left(6 t^{2} + 3 t + 6\right) x + 3$ |
$[1, 0]$ |
$[1]$ |
| 3.9.9.5 |
$9$ |
$x^{9} - 24 x^{7} + 9 x^{6} + 216 x^{5} - 144 x^{4} + 648 x^{2} - 216 x + 27$ |
$3$ |
$3$ |
$3$ |
$9$ |
$(C_3^3:C_3):C_2$ (as 9T22) |
$3$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2, 3/2]_{2}^{3}$ |
$t^{3} + 2 t + 1$ |
$x^{3} + \left(6 t^{2} + 3 t\right) x + 3$ |
$[1, 0]$ |
$[1]$ |
| 3.9.9.6 |
$9$ |
$x^{9} - 6 x^{8} + 45 x^{7} + 594 x^{6} + 99 x^{5} + 108 x^{4} - 54 x^{3} + 27 x^{2} + 81 x + 27$ |
$3$ |
$3$ |
$3$ |
$9$ |
$S_3\times C_3$ (as 9T4) |
$3$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{3}$ |
$t^{3} + 2 t + 1$ |
$x^{3} + \left(6 t^{2} + 3 t + 6\right) x^{2} + 3 x + 3$ |
$[1, 0]$ |
$[1]$ |
| 3.9.9.7 |
$9$ |
$x^{9} - 6 x^{7} + 9 x^{6} - 36 x^{5} - 36 x^{4} + 459 x^{3} - 108 x^{2} - 54 x + 27$ |
$3$ |
$3$ |
$3$ |
$9$ |
$C_3^2 : S_3 $ (as 9T13) |
$3$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{3}$ |
$t^{3} + 2 t + 1$ |
$x^{3} + \left(6 t^{2} + 6\right) x + 3$ |
$[1, 0]$ |
$[1]$ |
| 3.9.9.8 |
$9$ |
$x^{9} + 6 x^{8} + 114 x^{7} + 549 x^{6} + 522 x^{5} + 495 x^{4} + 216 x^{3} + 297 x^{2} - 108 x + 27$ |
$3$ |
$3$ |
$3$ |
$9$ |
$(C_3^3:C_3):C_2$ (as 9T22) |
$3$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2, 3/2]_{2}^{3}$ |
$t^{3} + 2 t + 1$ |
$x^{3} + \left(3 t^{2} + 6 t + 6\right) x^{2} + \left(3 t^{2} + 3 t\right) x + 3$ |
$[1, 0]$ |
$[1]$ |
| 3.9.9.9 |
$9$ |
$x^{9} + 9 x^{6} + 72 x^{5} - 189 x^{3} + 216 x^{2} + 27$ |
$3$ |
$3$ |
$3$ |
$9$ |
$(C_3^3:C_3):C_2$ (as 9T22) |
$3$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2, 3/2]_{2}^{3}$ |
$t^{3} + 2 t + 1$ |
$x^{3} + 6 t x + 3$ |
$[1, 0]$ |
$[1]$ |
