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Label Polynomial $p$ $e$ $f$ $c$ Galois group Visible slopes Slope content Unram. Ext. Eisen. Poly.
5.10.0.1 $x^{10} + 3 x^{5} + 3 x^{4} + 2 x^{3} + 4 x^{2} + x + 2$ $5$ $1$ $10$ $0$ $C_{10}$ (as 10T1) $[\ ]$ $[\ ]^{10}$ $t^{10} + 3 t^{5} + 3 t^{4} + 2 t^{3} + 4 t^{2} + t + 2$ $x - 5$
5.10.5.1 $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ $5$ $2$ $5$ $5$ $C_{10}$ (as 10T1) $[\ ]$ $[\ ]_{2}^{5}$ $t^{5} + 4 t + 3$ $x^{2} + 20 x + 5$
5.10.5.2 $x^{10} + 2500 x^{2} - 9375$ $5$ $2$ $5$ $5$ $C_{10}$ (as 10T1) $[\ ]$ $[\ ]_{2}^{5}$ $t^{5} + 4 t + 3$ $x^{2} + 5 t$
5.10.10.1 $x^{10} + 10 x^{5} - 50 x^{2} + 25$ $5$ $5$ $2$ $10$ $C_5^2 : C_8$ (as 10T18) $[5/4]$ $[5/4, 5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(5 t + 10\right) x + 5$
5.10.10.10 $x^{10} + 40 x^{6} + 10 x^{5} - 200 x^{4} + 400 x^{2} + 200 x + 25$ $5$ $5$ $2$ $10$ $F_{5}\times C_2$ (as 10T5) $[5/4]$ $[5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(10 t + 20\right) x^{2} + 20 x + 5$
5.10.10.11 $x^{10} - 20 x^{6} + 10 x^{5} - 100 x^{2} - 100 x + 25$ $5$ $5$ $2$ $10$ $(C_5^2 : C_4) : C_2$ (as 10T17) $[5/4]$ $[5/4, 5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(10 t + 10\right) x + 5$
5.10.10.12 $x^{10} - 30 x^{6} + 10 x^{5} + 25 x^{2} - 150 x + 25$ $5$ $5$ $2$ $10$ $(C_5^2 : C_4) : C_2$ (as 10T17) $[5/4]$ $[5/4, 5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(10 t + 5\right) x + 5$
5.10.10.13 $x^{10} - 10 x^{6} + 10 x^{5} - 25 x^{2} - 50 x + 25$ $5$ $5$ $2$ $10$ $(C_5^2 : C_4) : C_2$ (as 10T17) $[5/4]$ $[5/4, 5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(5 t + 5\right) x + 5$
5.10.10.14 $x^{10} + 10 x^{6} + 10 x^{5} - 25 x^{2} + 50 x + 25$ $5$ $5$ $2$ $10$ $(C_5^2 : C_4) : C_2$ (as 10T17) $[5/4]$ $[5/4, 5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(5 t + 15\right) x + 5$
5.10.10.15 $x^{10} + 5 x + 5$ $5$ $10$ $1$ $10$ $(C_5^2 : C_8):C_2$ (as 10T28) $[9/8]$ $[9/8, 9/8]_{8}^{2}$ $t + 3$ $x^{10} + 5 x + 5$
5.10.10.16 $x^{10} + 10 x + 5$ $5$ $10$ $1$ $10$ $(C_5^2 : C_8):C_2$ (as 10T28) $[9/8]$ $[9/8, 9/8]_{8}^{2}$ $t + 3$ $x^{10} + 10 x + 5$
5.10.10.17 $x^{10} + 5 x + 10$ $5$ $10$ $1$ $10$ $(C_5^2 : C_8):C_2$ (as 10T28) $[9/8]$ $[9/8, 9/8]_{8}^{2}$ $t + 3$ $x^{10} + 5 x + 10$
5.10.10.18 $x^{10} + 10 x + 10$ $5$ $10$ $1$ $10$ $(C_5^2 : C_8):C_2$ (as 10T28) $[9/8]$ $[9/8, 9/8]_{8}^{2}$ $t + 3$ $x^{10} + 10 x + 10$
5.10.10.2 $x^{10} - 20 x^{6} + 10 x^{5} + 50 x^{2} - 100 x + 25$ $5$ $5$ $2$ $10$ $C_5^2 : C_8$ (as 10T18) $[5/4]$ $[5/4, 5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + 5 t x + 5$
5.10.10.3 $x^{10} + 10 x^{5} - 200 x^{2} + 25$ $5$ $5$ $2$ $10$ $C_5^2 : C_8$ (as 10T18) $[5/4]$ $[5/4, 5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(10 t + 20\right) x + 5$
5.10.10.4 $x^{10} + 20 x^{6} + 10 x^{5} + 50 x^{2} + 100 x + 25$ $5$ $5$ $2$ $10$ $C_5^2 : C_8$ (as 10T18) $[5/4]$ $[5/4, 5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(5 t + 20\right) x + 5$
5.10.10.5 $x^{10} - 10 x^{6} + 10 x^{5} - 175 x^{2} - 50 x + 25$ $5$ $5$ $2$ $10$ $C_5^2 : C_8$ (as 10T18) $[5/4]$ $[5/4, 5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(10 t + 15\right) x + 5$
5.10.10.6 $x^{10} - 40 x^{6} + 10 x^{5} + 200 x^{2} - 200 x + 25$ $5$ $5$ $2$ $10$ $C_5^2 : C_8$ (as 10T18) $[5/4]$ $[5/4, 5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + 10 t x + 5$
5.10.10.7 $x^{10} - 50 x^{7} + 10 x^{6} + 10 x^{5} + 175 x^{4} - 250 x^{3} - 225 x^{2} + 50 x + 25$ $5$ $5$ $2$ $10$ $F_{5}\times C_2$ (as 10T5) $[5/4]$ $[5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(15 t + 5\right) x^{2} + 5 x + 5$
5.10.10.8 $x^{10} - 50 x^{7} + 30 x^{6} + 10 x^{5} - 175 x^{4} - 750 x^{3} - 25 x^{2} + 150 x + 25$ $5$ $5$ $2$ $10$ $F_{5}\times C_2$ (as 10T5) $[5/4]$ $[5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(20 t + 15\right) x^{2} + 15 x + 5$
5.10.10.9 $x^{10} + 20 x^{6} + 10 x^{5} - 50 x^{4} + 100 x^{2} + 100 x + 25$ $5$ $5$ $2$ $10$ $F_{5}\times C_2$ (as 10T5) $[5/4]$ $[5/4]_{4}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(5 t + 10\right) x^{2} + 10 x + 5$
5.10.11.1 $x^{10} + 20 x^{2} + 5$ $5$ $10$ $1$ $11$ $F_5$ (as 10T4) $[5/4]$ $[5/4]_{4}$ $t + 3$ $x^{10} + 20 x^{2} + 5$
5.10.11.2 $x^{10} + 5 x^{2} + 5$ $5$ $10$ $1$ $11$ $F_5$ (as 10T4) $[5/4]$ $[5/4]_{4}$ $t + 3$ $x^{10} + 5 x^{2} + 5$
5.10.11.3 $x^{10} + 10 x^{2} + 5$ $5$ $10$ $1$ $11$ $F_{5}\times C_2$ (as 10T5) $[5/4]$ $[5/4]_{4}^{2}$ $t + 3$ $x^{10} + 10 x^{2} + 5$
5.10.11.4 $x^{10} + 15 x^{2} + 5$ $5$ $10$ $1$ $11$ $F_{5}\times C_2$ (as 10T5) $[5/4]$ $[5/4]_{4}^{2}$ $t + 3$ $x^{10} + 15 x^{2} + 5$
5.10.11.5 $x^{10} + 15 x^{2} + 10$ $5$ $10$ $1$ $11$ $F_5$ (as 10T4) $[5/4]$ $[5/4]_{4}$ $t + 3$ $x^{10} + 15 x^{2} + 10$
5.10.11.6 $x^{10} + 10 x^{2} + 10$ $5$ $10$ $1$ $11$ $F_5$ (as 10T4) $[5/4]$ $[5/4]_{4}$ $t + 3$ $x^{10} + 10 x^{2} + 10$
5.10.11.7 $x^{10} + 5 x^{2} + 10$ $5$ $10$ $1$ $11$ $F_{5}\times C_2$ (as 10T5) $[5/4]$ $[5/4]_{4}^{2}$ $t + 3$ $x^{10} + 5 x^{2} + 10$
5.10.11.8 $x^{10} + 20 x^{2} + 10$ $5$ $10$ $1$ $11$ $F_{5}\times C_2$ (as 10T5) $[5/4]$ $[5/4]_{4}^{2}$ $t + 3$ $x^{10} + 20 x^{2} + 10$
5.10.12.1 $x^{10} + 10 x^{5} - 50 x^{4} + 25$ $5$ $5$ $2$ $12$ $C_5^2 : C_8$ (as 10T18) $[3/2]$ $[3/2, 3/2]_{2}^{4}$ $t^{2} + 4 t + 2$ $x^{5} + \left(5 t + 10\right) x^{2} + 5$
5.10.12.10 $x^{10} + 20 x^{7} - 200 x^{6} + 10 x^{5} + 100 x^{4} + 100 x^{2} + 25$ $5$ $5$ $2$ $12$ $D_{10}$ (as 10T3) $[3/2]$ $[3/2]_{2}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(10 t + 20\right) x^{3} + 10 x^{2} + 5$
5.10.12.11 $x^{10} - 20 x^{7} + 10 x^{5} - 100 x^{4} - 100 x^{2} + 25$ $5$ $5$ $2$ $12$ $C_5^2 : C_4$ (as 10T10) $[3/2]$ $[3/2, 3/2]_{2}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(10 t + 10\right) x^{2} + 5$
5.10.12.12 $x^{10} - 30 x^{7} + 10 x^{5} + 25 x^{4} - 150 x^{2} + 25$ $5$ $5$ $2$ $12$ $C_5^2 : C_4$ (as 10T10) $[3/2]$ $[3/2, 3/2]_{2}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(10 t + 5\right) x^{2} + 5$
5.10.12.13 $x^{10} - 10 x^{7} + 10 x^{5} - 25 x^{4} - 50 x^{2} + 25$ $5$ $5$ $2$ $12$ $D_5^2$ (as 10T9) $[3/2]$ $[3/2, 3/2]_{2}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(5 t + 5\right) x^{2} + 5$
5.10.12.14 $x^{10} + 10 x^{7} + 10 x^{5} - 25 x^{4} + 50 x^{2} + 25$ $5$ $5$ $2$ $12$ $D_5^2$ (as 10T9) $[3/2]$ $[3/2, 3/2]_{2}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(5 t + 15\right) x^{2} + 5$
5.10.12.15 $x^{10} + 5 x^{3} + 5$ $5$ $10$ $1$ $12$ $(C_5^2 : C_8):C_2$ (as 10T28) $[11/8]$ $[11/8, 11/8]_{8}^{2}$ $t + 3$ $x^{10} + 5 x^{3} + 5$
5.10.12.16 $x^{10} + 10 x^{3} + 10$ $5$ $10$ $1$ $12$ $(C_5^2 : C_8):C_2$ (as 10T28) $[11/8]$ $[11/8, 11/8]_{8}^{2}$ $t + 3$ $x^{10} + 10 x^{3} + 10$
5.10.12.17 $x^{10} + 10 x^{3} + 5$ $5$ $10$ $1$ $12$ $(C_5^2 : C_8):C_2$ (as 10T28) $[11/8]$ $[11/8, 11/8]_{8}^{2}$ $t + 3$ $x^{10} + 10 x^{3} + 5$
5.10.12.18 $x^{10} + 5 x^{3} + 10$ $5$ $10$ $1$ $12$ $(C_5^2 : C_8):C_2$ (as 10T28) $[11/8]$ $[11/8, 11/8]_{8}^{2}$ $t + 3$ $x^{10} + 5 x^{3} + 10$
5.10.12.2 $x^{10} - 10 x^{7} + 10 x^{5} - 175 x^{4} - 50 x^{2} + 25$ $5$ $5$ $2$ $12$ $C_5^2 : C_8$ (as 10T18) $[3/2]$ $[3/2, 3/2]_{2}^{4}$ $t^{2} + 4 t + 2$ $x^{5} + \left(10 t + 15\right) x^{2} + 5$
5.10.12.3 $x^{10} - 20 x^{7} + 10 x^{5} + 50 x^{4} - 100 x^{2} + 25$ $5$ $5$ $2$ $12$ $C_5^2 : C_8$ (as 10T18) $[3/2]$ $[3/2, 3/2]_{2}^{4}$ $t^{2} + 4 t + 2$ $x^{5} + 5 t x^{2} + 5$
5.10.12.4 $x^{10} + 10 x^{5} - 200 x^{4} + 25$ $5$ $5$ $2$ $12$ $C_5^2 : C_8$ (as 10T18) $[3/2]$ $[3/2, 3/2]_{2}^{4}$ $t^{2} + 4 t + 2$ $x^{5} + \left(10 t + 20\right) x^{2} + 5$
5.10.12.5 $x^{10} + 20 x^{7} + 10 x^{5} + 50 x^{4} + 100 x^{2} + 25$ $5$ $5$ $2$ $12$ $C_5^2 : C_8$ (as 10T18) $[3/2]$ $[3/2, 3/2]_{2}^{4}$ $t^{2} + 4 t + 2$ $x^{5} + \left(5 t + 20\right) x^{2} + 5$
5.10.12.6 $x^{10} - 40 x^{7} + 10 x^{5} + 200 x^{4} - 200 x^{2} + 25$ $5$ $5$ $2$ $12$ $C_5^2 : C_8$ (as 10T18) $[3/2]$ $[3/2, 3/2]_{2}^{4}$ $t^{2} + 4 t + 2$ $x^{5} + 10 t x^{2} + 5$
5.10.12.7 $x^{10} - 50 x^{8} + 30 x^{7} + 175 x^{6} - 740 x^{5} + 225 x^{4} - 250 x^{3} + 150 x^{2} + 25$ $5$ $5$ $2$ $12$ $D_{10}$ (as 10T3) $[3/2]$ $[3/2]_{2}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(15 t + 5\right) x^{3} + 15 x^{2} + 5$
5.10.12.8 $x^{10} - 50 x^{8} + 40 x^{7} - 175 x^{6} - 990 x^{5} + 400 x^{4} - 250 x^{3} + 200 x^{2} + 25$ $5$ $5$ $2$ $12$ $F_5$ (as 10T4) $[3/2]$ $[3/2]_{2}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(20 t + 15\right) x^{3} + 20 x^{2} + 5$
5.10.12.9 $x^{10} + 10 x^{7} - 50 x^{6} + 10 x^{5} + 25 x^{4} + 50 x^{2} + 25$ $5$ $5$ $2$ $12$ $F_5$ (as 10T4) $[3/2]$ $[3/2]_{2}^{2}$ $t^{2} + 4 t + 2$ $x^{5} + \left(5 t + 10\right) x^{3} + 5 x^{2} + 5$
5.10.13.1 $x^{10} + 15 x^{4} + 5$ $5$ $10$ $1$ $13$ $D_5$ (as 10T2) $[3/2]$ $[3/2]_{2}$ $t + 3$ $x^{10} + 15 x^{4} + 5$
5.10.13.10 $x^{10} + 10 x^{4} + 10$ $5$ $10$ $1$ $13$ $F_{5}\times C_2$ (as 10T5) $[3/2]$ $[3/2]_{2}^{4}$ $t + 3$ $x^{10} + 10 x^{4} + 10$
5.10.13.11 $x^{10} + 10 x^{5} + 5 x^{4} + 10$ $5$ $10$ $1$ $13$ $D_5\times C_5$ (as 10T6) $[3/2]$ $[3/2]_{2}^{5}$ $t + 3$ $x^{10} + 10 x^{5} + 5 x^{4} + 10$
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