Label |
$n$ |
Polynomial |
$p$ |
$e$ |
$f$ |
$c$ |
Galois group |
$u$ |
$t$ |
Visible slopes |
Slope content |
Ind. of Insep. |
Assoc. Inertia |
5.10.0.1 |
$10$ |
x10 + 3x5 + 3x4 + 2x3 + 4x2 + x + 2 |
$5$ |
$1$ |
$10$ |
$0$ |
$C_{10}$ (as 10T1) |
$10$ |
$1$ |
$[\ ]$ |
$[\ ]^{10}$ |
$[0]$ |
$[]$ |
5.10.5.1 |
$10$ |
x10 + 100x9 + 4025x8 + 82000x7 + 860258x6 + 4015486x5 + 4317350x4 + 2373700x3 + 3853141x2 + 15123594x + 12051954 |
$5$ |
$2$ |
$5$ |
$5$ |
$C_{10}$ (as 10T1) |
$5$ |
$2$ |
$[\ ]$ |
$[\ ]_{2}^{5}$ |
$[0]$ |
$[1]$ |
5.10.5.2 |
$10$ |
x10 + 2500x2 - 9375 |
$5$ |
$2$ |
$5$ |
$5$ |
$C_{10}$ (as 10T1) |
$5$ |
$2$ |
$[\ ]$ |
$[\ ]_{2}^{5}$ |
$[0]$ |
$[1]$ |
5.10.10.1 |
$10$ |
x10 + 10x5 - 50x2 + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$C_5^2 : C_8$ (as 10T18) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4, 5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.10.10 |
$10$ |
x10 + 40x6 + 10x5 - 200x4 + 400x2 + 200x + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$F_{5}\times C_2$ (as 10T5) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.10.11 |
$10$ |
x10 - 20x6 + 10x5 - 100x2 - 100x + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$(C_5^2 : C_4) : C_2$ (as 10T17) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4, 5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.10.12 |
$10$ |
x10 - 30x6 + 10x5 + 25x2 - 150x + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$(C_5^2 : C_4) : C_2$ (as 10T17) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4, 5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.10.13 |
$10$ |
x10 - 10x6 + 10x5 - 25x2 - 50x + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$(C_5^2 : C_4) : C_2$ (as 10T17) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4, 5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.10.14 |
$10$ |
x10 + 10x6 + 10x5 - 25x2 + 50x + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$(C_5^2 : C_4) : C_2$ (as 10T17) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4, 5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.10.15 |
$10$ |
x10 + 5x + 5 |
$5$ |
$10$ |
$1$ |
$10$ |
$(C_5^2 : C_8):C_2$ (as 10T28) |
$2$ |
$8$ |
$[9/8]$ |
$[9/8, 9/8]_{8}^{2}$ |
$[1, 0]$ |
$[1, 1]$ |
5.10.10.16 |
$10$ |
x10 + 10x + 5 |
$5$ |
$10$ |
$1$ |
$10$ |
$(C_5^2 : C_8):C_2$ (as 10T28) |
$2$ |
$8$ |
$[9/8]$ |
$[9/8, 9/8]_{8}^{2}$ |
$[1, 0]$ |
$[1, 1]$ |
5.10.10.17 |
$10$ |
x10 + 5x + 10 |
$5$ |
$10$ |
$1$ |
$10$ |
$(C_5^2 : C_8):C_2$ (as 10T28) |
$2$ |
$8$ |
$[9/8]$ |
$[9/8, 9/8]_{8}^{2}$ |
$[1, 0]$ |
$[1, 1]$ |
5.10.10.18 |
$10$ |
x10 + 10x + 10 |
$5$ |
$10$ |
$1$ |
$10$ |
$(C_5^2 : C_8):C_2$ (as 10T28) |
$2$ |
$8$ |
$[9/8]$ |
$[9/8, 9/8]_{8}^{2}$ |
$[1, 0]$ |
$[1, 1]$ |
5.10.10.2 |
$10$ |
x10 - 20x6 + 10x5 + 50x2 - 100x + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$C_5^2 : C_8$ (as 10T18) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4, 5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.10.3 |
$10$ |
x10 + 10x5 - 200x2 + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$C_5^2 : C_8$ (as 10T18) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4, 5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.10.4 |
$10$ |
x10 + 20x6 + 10x5 + 50x2 + 100x + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$C_5^2 : C_8$ (as 10T18) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4, 5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.10.5 |
$10$ |
x10 - 10x6 + 10x5 - 175x2 - 50x + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$C_5^2 : C_8$ (as 10T18) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4, 5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.10.6 |
$10$ |
x10 - 40x6 + 10x5 + 200x2 - 200x + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$C_5^2 : C_8$ (as 10T18) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4, 5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.10.7 |
$10$ |
x10 - 50x7 + 10x6 + 10x5 + 175x4 - 250x3 - 225x2 + 50x + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$F_{5}\times C_2$ (as 10T5) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.10.8 |
$10$ |
x10 - 50x7 + 30x6 + 10x5 - 175x4 - 750x3 - 25x2 + 150x + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$F_{5}\times C_2$ (as 10T5) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.10.9 |
$10$ |
x10 + 20x6 + 10x5 - 50x4 + 100x2 + 100x + 25 |
$5$ |
$5$ |
$2$ |
$10$ |
$F_{5}\times C_2$ (as 10T5) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4]_{4}^{2}$ |
$[1, 0]$ |
$[1]$ |
5.10.11.1 |
$10$ |
x10 + 20x2 + 5 |
$5$ |
$10$ |
$1$ |
$11$ |
$F_5$ (as 10T4) |
$1$ |
$4$ |
$[5/4]$ |
$[5/4]_{4}$ |
$[2, 0]$ |
$[1, 1]$ |
5.10.11.2 |
$10$ |
x10 + 5x2 + 5 |
$5$ |
$10$ |
$1$ |
$11$ |
$F_5$ (as 10T4) |
$1$ |
$4$ |
$[5/4]$ |
$[5/4]_{4}$ |
$[2, 0]$ |
$[1, 1]$ |
5.10.11.3 |
$10$ |
x10 + 10x2 + 5 |
$5$ |
$10$ |
$1$ |
$11$ |
$F_{5}\times C_2$ (as 10T5) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4]_{4}^{2}$ |
$[2, 0]$ |
$[2, 1]$ |
5.10.11.4 |
$10$ |
x10 + 15x2 + 5 |
$5$ |
$10$ |
$1$ |
$11$ |
$F_{5}\times C_2$ (as 10T5) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4]_{4}^{2}$ |
$[2, 0]$ |
$[2, 1]$ |
5.10.11.5 |
$10$ |
x10 + 15x2 + 10 |
$5$ |
$10$ |
$1$ |
$11$ |
$F_5$ (as 10T4) |
$1$ |
$4$ |
$[5/4]$ |
$[5/4]_{4}$ |
$[2, 0]$ |
$[1, 1]$ |
5.10.11.6 |
$10$ |
x10 + 10x2 + 10 |
$5$ |
$10$ |
$1$ |
$11$ |
$F_5$ (as 10T4) |
$1$ |
$4$ |
$[5/4]$ |
$[5/4]_{4}$ |
$[2, 0]$ |
$[1, 1]$ |
5.10.11.7 |
$10$ |
x10 + 5x2 + 10 |
$5$ |
$10$ |
$1$ |
$11$ |
$F_{5}\times C_2$ (as 10T5) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4]_{4}^{2}$ |
$[2, 0]$ |
$[2, 1]$ |
5.10.11.8 |
$10$ |
x10 + 20x2 + 10 |
$5$ |
$10$ |
$1$ |
$11$ |
$F_{5}\times C_2$ (as 10T5) |
$2$ |
$4$ |
$[5/4]$ |
$[5/4]_{4}^{2}$ |
$[2, 0]$ |
$[2, 1]$ |
5.10.12.1 |
$10$ |
x10 + 10x5 - 50x4 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$C_5^2 : C_8$ (as 10T18) |
$4$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{4}$ |
$[2, 0]$ |
$[2]$ |
5.10.12.10 |
$10$ |
x10 + 20x7 - 200x6 + 10x5 + 100x4 + 100x2 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$D_{10}$ (as 10T3) |
$2$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{2}$ |
$[2, 0]$ |
$[1]$ |
5.10.12.11 |
$10$ |
x10 - 20x7 + 10x5 - 100x4 - 100x2 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$C_5^2 : C_4$ (as 10T10) |
$2$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{2}$ |
$[2, 0]$ |
$[1]$ |
5.10.12.12 |
$10$ |
x10 - 30x7 + 10x5 + 25x4 - 150x2 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$C_5^2 : C_4$ (as 10T10) |
$2$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{2}$ |
$[2, 0]$ |
$[1]$ |
5.10.12.13 |
$10$ |
x10 - 10x7 + 10x5 - 25x4 - 50x2 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$D_5^2$ (as 10T9) |
$2$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{2}$ |
$[2, 0]$ |
$[1]$ |
5.10.12.14 |
$10$ |
x10 + 10x7 + 10x5 - 25x4 + 50x2 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$D_5^2$ (as 10T9) |
$2$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{2}$ |
$[2, 0]$ |
$[1]$ |
5.10.12.15 |
$10$ |
x10 + 5x3 + 5 |
$5$ |
$10$ |
$1$ |
$12$ |
$(C_5^2 : C_8):C_2$ (as 10T28) |
$2$ |
$8$ |
$[11/8]$ |
$[11/8, 11/8]_{8}^{2}$ |
$[3, 0]$ |
$[1, 1]$ |
5.10.12.16 |
$10$ |
x10 + 10x3 + 10 |
$5$ |
$10$ |
$1$ |
$12$ |
$(C_5^2 : C_8):C_2$ (as 10T28) |
$2$ |
$8$ |
$[11/8]$ |
$[11/8, 11/8]_{8}^{2}$ |
$[3, 0]$ |
$[1, 1]$ |
5.10.12.17 |
$10$ |
x10 + 10x3 + 5 |
$5$ |
$10$ |
$1$ |
$12$ |
$(C_5^2 : C_8):C_2$ (as 10T28) |
$2$ |
$8$ |
$[11/8]$ |
$[11/8, 11/8]_{8}^{2}$ |
$[3, 0]$ |
$[1, 1]$ |
5.10.12.18 |
$10$ |
x10 + 5x3 + 10 |
$5$ |
$10$ |
$1$ |
$12$ |
$(C_5^2 : C_8):C_2$ (as 10T28) |
$2$ |
$8$ |
$[11/8]$ |
$[11/8, 11/8]_{8}^{2}$ |
$[3, 0]$ |
$[1, 1]$ |
5.10.12.2 |
$10$ |
x10 - 10x7 + 10x5 - 175x4 - 50x2 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$C_5^2 : C_8$ (as 10T18) |
$4$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{4}$ |
$[2, 0]$ |
$[2]$ |
5.10.12.3 |
$10$ |
x10 - 20x7 + 10x5 + 50x4 - 100x2 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$C_5^2 : C_8$ (as 10T18) |
$4$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{4}$ |
$[2, 0]$ |
$[2]$ |
5.10.12.4 |
$10$ |
x10 + 10x5 - 200x4 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$C_5^2 : C_8$ (as 10T18) |
$4$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{4}$ |
$[2, 0]$ |
$[2]$ |
5.10.12.5 |
$10$ |
x10 + 20x7 + 10x5 + 50x4 + 100x2 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$C_5^2 : C_8$ (as 10T18) |
$4$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{4}$ |
$[2, 0]$ |
$[2]$ |
5.10.12.6 |
$10$ |
x10 - 40x7 + 10x5 + 200x4 - 200x2 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$C_5^2 : C_8$ (as 10T18) |
$4$ |
$2$ |
$[3/2]$ |
$[3/2, 3/2]_{2}^{4}$ |
$[2, 0]$ |
$[2]$ |
5.10.12.7 |
$10$ |
x10 - 50x8 + 30x7 + 175x6 - 740x5 + 225x4 - 250x3 + 150x2 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$D_{10}$ (as 10T3) |
$2$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{2}$ |
$[2, 0]$ |
$[1]$ |
5.10.12.8 |
$10$ |
x10 - 50x8 + 40x7 - 175x6 - 990x5 + 400x4 - 250x3 + 200x2 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$F_5$ (as 10T4) |
$2$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{2}$ |
$[2, 0]$ |
$[1]$ |
5.10.12.9 |
$10$ |
x10 + 10x7 - 50x6 + 10x5 + 25x4 + 50x2 + 25 |
$5$ |
$5$ |
$2$ |
$12$ |
$F_5$ (as 10T4) |
$2$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{2}$ |
$[2, 0]$ |
$[1]$ |
5.10.13.1 |
$10$ |
x10 + 15x4 + 5 |
$5$ |
$10$ |
$1$ |
$13$ |
$D_5$ (as 10T2) |
$1$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}$ |
$[4, 0]$ |
$[1, 1]$ |
5.10.13.10 |
$10$ |
x10 + 10x4 + 10 |
$5$ |
$10$ |
$1$ |
$13$ |
$F_{5}\times C_2$ (as 10T5) |
$4$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{4}$ |
$[4, 0]$ |
$[4, 1]$ |
5.10.13.11 |
$10$ |
x10 + 10x5 + 5x4 + 10 |
$5$ |
$10$ |
$1$ |
$13$ |
$D_5\times C_5$ (as 10T6) |
$5$ |
$2$ |
$[3/2]$ |
$[3/2]_{2}^{5}$ |
$[4, 0]$ |
$[1, 1]$ |