Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Ind. of Insep. |
Assoc. Inertia |
2.4.0.1 |
$4$ |
x4 + x + 1 |
$2$ |
$1$ |
$4$ |
$0$ |
$C_4$ (as 4T1) |
$4$ |
$1$ |
$[\ ]$ |
$[\ ]^{4}$ |
$[0]$ |
$[]$ |
2.4.4.1 |
$4$ |
x4 + 6x3 + 17x2 + 24x + 13 |
$2$ |
$2$ |
$2$ |
$4$ |
$C_2^2$ (as 4T2) |
$2$ |
$1$ |
$[2]$ |
$[2]^{2}$ |
$[1, 0]$ |
$[1]$ |
2.4.4.2 |
$4$ |
x4 + 4x3 + 4x2 + 12 |
$2$ |
$2$ |
$2$ |
$4$ |
$C_4$ (as 4T1) |
$2$ |
$1$ |
$[2]$ |
$[2]^{2}$ |
$[1, 0]$ |
$[1]$ |
2.4.4.3 |
$4$ |
x4 + 2x3 + 4x2 + 12x + 12 |
$2$ |
$2$ |
$2$ |
$4$ |
$D_{4}$ (as 4T3) |
$2$ |
$1$ |
$[2]$ |
$[2, 2]^{2}$ |
$[1, 0]$ |
$[1]$ |
2.4.4.4 |
$4$ |
x4 - 2x3 + 4x2 + 12x + 12 |
$2$ |
$2$ |
$2$ |
$4$ |
$D_{4}$ (as 4T3) |
$2$ |
$1$ |
$[2]$ |
$[2, 2]^{2}$ |
$[1, 0]$ |
$[1]$ |
2.4.4.5 |
$4$ |
x4 + 2x + 2 |
$2$ |
$4$ |
$1$ |
$4$ |
$S_4$ (as 4T5) |
$2$ |
$3$ |
$[4/3, 4/3]$ |
$[4/3, 4/3]_{3}^{2}$ |
$[1, 1, 0]$ |
$[1]$ |
2.4.6.1 |
$4$ |
x4 + 2x3 + 31x2 + 30x + 183 |
$2$ |
$2$ |
$2$ |
$6$ |
$C_2^2$ (as 4T2) |
$2$ |
$1$ |
$[3]$ |
$[3]^{2}$ |
$[2, 0]$ |
$[1]$ |
2.4.6.2 |
$4$ |
x4 + 4x3 + 16x2 + 24x + 12 |
$2$ |
$2$ |
$2$ |
$6$ |
$C_2^2$ (as 4T2) |
$2$ |
$1$ |
$[3]$ |
$[3]^{2}$ |
$[2, 0]$ |
$[1]$ |
2.4.6.3 |
$4$ |
x4 + 8x3 + 28x2 + 48x + 84 |
$2$ |
$2$ |
$2$ |
$6$ |
$C_4$ (as 4T1) |
$2$ |
$1$ |
$[3]$ |
$[3]^{2}$ |
$[2, 0]$ |
$[1]$ |
2.4.6.4 |
$4$ |
x4 + 4x3 + 24x2 + 88x + 124 |
$2$ |
$2$ |
$2$ |
$6$ |
$C_4$ (as 4T1) |
$2$ |
$1$ |
$[3]$ |
$[3]^{2}$ |
$[2, 0]$ |
$[1]$ |
2.4.6.5 |
$4$ |
x4 - 4x3 + 36x2 + 8x + 148 |
$2$ |
$2$ |
$2$ |
$6$ |
$D_{4}$ (as 4T3) |
$2$ |
$1$ |
$[3]$ |
$[2, 3]^{2}$ |
$[2, 0]$ |
$[1]$ |
2.4.6.6 |
$4$ |
x4 - 4x3 + 28x2 - 24x + 36 |
$2$ |
$2$ |
$2$ |
$6$ |
$D_{4}$ (as 4T3) |
$2$ |
$1$ |
$[3]$ |
$[2, 3]^{2}$ |
$[2, 0]$ |
$[1]$ |
2.4.6.7 |
$4$ |
x4 + 2x3 + 2x2 + 2 |
$2$ |
$4$ |
$1$ |
$6$ |
$A_4$ (as 4T4) |
$3$ |
$1$ |
$[2, 2]$ |
$[2, 2]^{3}$ |
$[3, 2, 0]$ |
$[3]$ |
2.4.6.8 |
$4$ |
x4 + 2x3 + 2 |
$2$ |
$4$ |
$1$ |
$6$ |
$D_{4}$ (as 4T3) |
$2$ |
$1$ |
$[2, 2]$ |
$[2, 2]^{2}$ |
$[3, 3, 0]$ |
$[2]$ |
2.4.6.9 |
$4$ |
x4 + 2x3 + 6 |
$2$ |
$4$ |
$1$ |
$6$ |
$D_{4}$ (as 4T3) |
$2$ |
$1$ |
$[2, 2]$ |
$[2, 2]^{2}$ |
$[3, 3, 0]$ |
$[2]$ |
2.4.8.1 |
$4$ |
x4 + 2x2 + 4x + 10 |
$2$ |
$4$ |
$1$ |
$8$ |
$C_2^2$ (as 4T2) |
$1$ |
$1$ |
$[2, 3]$ |
$[2, 3]$ |
$[5, 2, 0]$ |
$[1, 1]$ |
2.4.8.2 |
$4$ |
x4 + 2x2 + 4x + 2 |
$2$ |
$4$ |
$1$ |
$8$ |
$C_2^2$ (as 4T2) |
$1$ |
$1$ |
$[2, 3]$ |
$[2, 3]$ |
$[5, 2, 0]$ |
$[1, 1]$ |
2.4.8.3 |
$4$ |
x4 + 6x2 + 4x + 14 |
$2$ |
$4$ |
$1$ |
$8$ |
$C_2^2$ (as 4T2) |
$1$ |
$1$ |
$[2, 3]$ |
$[2, 3]$ |
$[5, 2, 0]$ |
$[1, 1]$ |
2.4.8.4 |
$4$ |
x4 + 6x2 + 4x + 6 |
$2$ |
$4$ |
$1$ |
$8$ |
$C_2^2$ (as 4T2) |
$1$ |
$1$ |
$[2, 3]$ |
$[2, 3]$ |
$[5, 2, 0]$ |
$[1, 1]$ |
2.4.8.5 |
$4$ |
x4 + 2x2 + 4x + 6 |
$2$ |
$4$ |
$1$ |
$8$ |
$D_{4}$ (as 4T3) |
$2$ |
$1$ |
$[2, 3]$ |
$[2, 3]^{2}$ |
$[5, 2, 0]$ |
$[1, 1]$ |
2.4.8.6 |
$4$ |
x4 + 6x2 + 4x + 10 |
$2$ |
$4$ |
$1$ |
$8$ |
$D_{4}$ (as 4T3) |
$2$ |
$1$ |
$[2, 3]$ |
$[2, 3]^{2}$ |
$[5, 2, 0]$ |
$[1, 1]$ |
2.4.8.7 |
$4$ |
x4 + 4x2 + 4x + 2 |
$2$ |
$4$ |
$1$ |
$8$ |
$S_4$ (as 4T5) |
$2$ |
$3$ |
$[8/3, 8/3]$ |
$[8/3, 8/3]_{3}^{2}$ |
$[5, 4, 0]$ |
$[1]$ |
2.4.8.8 |
$4$ |
x4 + 4x + 2 |
$2$ |
$4$ |
$1$ |
$8$ |
$S_4$ (as 4T5) |
$2$ |
$3$ |
$[8/3, 8/3]$ |
$[8/3, 8/3]_{3}^{2}$ |
$[5, 4, 0]$ |
$[1]$ |
2.4.9.1 |
$4$ |
x4 + 10x2 + 2 |
$2$ |
$4$ |
$1$ |
$9$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[2, 7/2]$ |
$[2, 3, 7/2]$ |
$[6, 2, 0]$ |
$[1, 1]$ |
2.4.9.2 |
$4$ |
x4 + 2x2 + 2 |
$2$ |
$4$ |
$1$ |
$9$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[2, 7/2]$ |
$[2, 3, 7/2]$ |
$[6, 2, 0]$ |
$[1, 1]$ |
2.4.9.3 |
$4$ |
x4 + 4x3 + 2x2 + 2 |
$2$ |
$4$ |
$1$ |
$9$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[2, 7/2]$ |
$[2, 3, 7/2]$ |
$[6, 2, 0]$ |
$[1, 1]$ |
2.4.9.4 |
$4$ |
x4 + 10x2 + 8x + 2 |
$2$ |
$4$ |
$1$ |
$9$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[2, 7/2]$ |
$[2, 3, 7/2]$ |
$[6, 2, 0]$ |
$[1, 1]$ |
2.4.9.5 |
$4$ |
x4 + 10x2 + 8x + 6 |
$2$ |
$4$ |
$1$ |
$9$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[2, 7/2]$ |
$[2, 3, 7/2]$ |
$[6, 2, 0]$ |
$[1, 1]$ |
2.4.9.6 |
$4$ |
x4 + 4x3 + 10x2 + 14 |
$2$ |
$4$ |
$1$ |
$9$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[2, 7/2]$ |
$[2, 3, 7/2]$ |
$[6, 2, 0]$ |
$[1, 1]$ |
2.4.9.7 |
$4$ |
x4 + 2x2 + 6 |
$2$ |
$4$ |
$1$ |
$9$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[2, 7/2]$ |
$[2, 3, 7/2]$ |
$[6, 2, 0]$ |
$[1, 1]$ |
2.4.9.8 |
$4$ |
x4 + 10x2 + 6 |
$2$ |
$4$ |
$1$ |
$9$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[2, 7/2]$ |
$[2, 3, 7/2]$ |
$[6, 2, 0]$ |
$[1, 1]$ |
2.4.10.1 |
$4$ |
x4 + 4x3 + 10 |
$2$ |
$4$ |
$1$ |
$10$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[3, 7/2]$ |
$[2, 3, 7/2]$ |
$[7, 4, 0]$ |
$[1, 1]$ |
2.4.10.2 |
$4$ |
x4 + 4x3 + 2 |
$2$ |
$4$ |
$1$ |
$10$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[3, 7/2]$ |
$[2, 3, 7/2]$ |
$[7, 4, 0]$ |
$[1, 1]$ |
2.4.10.3 |
$4$ |
x4 + 4x3 + 8x2 + 2 |
$2$ |
$4$ |
$1$ |
$10$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[3, 7/2]$ |
$[2, 3, 7/2]$ |
$[7, 4, 0]$ |
$[1, 1]$ |
2.4.10.4 |
$4$ |
x4 + 4x3 + 8x2 + 10 |
$2$ |
$4$ |
$1$ |
$10$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[3, 7/2]$ |
$[2, 3, 7/2]$ |
$[7, 4, 0]$ |
$[1, 1]$ |
2.4.10.5 |
$4$ |
x4 + 4x3 + 4x2 + 10 |
$2$ |
$4$ |
$1$ |
$10$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[3, 7/2]$ |
$[2, 3, 7/2]$ |
$[7, 4, 0]$ |
$[1, 1]$ |
2.4.10.6 |
$4$ |
x4 + 4x3 + 12x2 + 10 |
$2$ |
$4$ |
$1$ |
$10$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[3, 7/2]$ |
$[2, 3, 7/2]$ |
$[7, 4, 0]$ |
$[1, 1]$ |
2.4.10.7 |
$4$ |
x4 + 4x3 + 4x2 + 2 |
$2$ |
$4$ |
$1$ |
$10$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[3, 7/2]$ |
$[2, 3, 7/2]$ |
$[7, 4, 0]$ |
$[1, 1]$ |
2.4.10.8 |
$4$ |
x4 + 4x3 + 12x2 + 2 |
$2$ |
$4$ |
$1$ |
$10$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[3, 7/2]$ |
$[2, 3, 7/2]$ |
$[7, 4, 0]$ |
$[1, 1]$ |
2.4.11.1 |
$4$ |
x4 + 8x3 + 4x2 + 2 |
$2$ |
$4$ |
$1$ |
$11$ |
$C_4$ (as 4T1) |
$1$ |
$1$ |
$[3, 4]$ |
$[3, 4]$ |
$[8, 4, 0]$ |
$[1, 1]$ |
2.4.11.10 |
$4$ |
x4 + 4x2 + 26 |
$2$ |
$4$ |
$1$ |
$11$ |
$C_4$ (as 4T1) |
$1$ |
$1$ |
$[3, 4]$ |
$[3, 4]$ |
$[8, 4, 0]$ |
$[1, 1]$ |
2.4.11.11 |
$4$ |
x4 + 10 |
$2$ |
$4$ |
$1$ |
$11$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[3, 4]$ |
$[2, 3, 4]$ |
$[8, 4, 0]$ |
$[1, 1]$ |
2.4.11.12 |
$4$ |
x4 + 26 |
$2$ |
$4$ |
$1$ |
$11$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[3, 4]$ |
$[2, 3, 4]$ |
$[8, 4, 0]$ |
$[1, 1]$ |
2.4.11.13 |
$4$ |
x4 + 8x3 + 8x + 10 |
$2$ |
$4$ |
$1$ |
$11$ |
$D_{4}$ (as 4T3) |
$2$ |
$1$ |
$[3, 4]$ |
$[3, 4]^{2}$ |
$[8, 4, 0]$ |
$[1, 1]$ |
2.4.11.14 |
$4$ |
x4 + 8x + 10 |
$2$ |
$4$ |
$1$ |
$11$ |
$D_{4}$ (as 4T3) |
$2$ |
$1$ |
$[3, 4]$ |
$[3, 4]^{2}$ |
$[8, 4, 0]$ |
$[1, 1]$ |
2.4.11.15 |
$4$ |
x4 + 12x2 + 8x + 2 |
$2$ |
$4$ |
$1$ |
$11$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[3, 4]$ |
$[2, 3, 4]$ |
$[8, 4, 0]$ |
$[1, 1]$ |
2.4.11.16 |
$4$ |
x4 + 12x2 + 8x + 18 |
$2$ |
$4$ |
$1$ |
$11$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[3, 4]$ |
$[2, 3, 4]$ |
$[8, 4, 0]$ |
$[1, 1]$ |
2.4.11.17 |
$4$ |
x4 + 8x3 + 8x + 2 |
$2$ |
$4$ |
$1$ |
$11$ |
$D_{4}$ (as 4T3) |
$2$ |
$1$ |
$[3, 4]$ |
$[3, 4]^{2}$ |
$[8, 4, 0]$ |
$[1, 1]$ |
2.4.11.18 |
$4$ |
x4 + 8x + 18 |
$2$ |
$4$ |
$1$ |
$11$ |
$D_{4}$ (as 4T3) |
$2$ |
$1$ |
$[3, 4]$ |
$[3, 4]^{2}$ |
$[8, 4, 0]$ |
$[1, 1]$ |
2.4.11.19 |
$4$ |
x4 + 12x2 + 8x + 26 |
$2$ |
$4$ |
$1$ |
$11$ |
$D_{4}$ (as 4T3) |
$1$ |
$1$ |
$[3, 4]$ |
$[2, 3, 4]$ |
$[8, 4, 0]$ |
$[1, 1]$ |