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Degree	Residue characteristic	Galois group	Galois unramified degree	Assoc. Inertia

Ramification index	Discriminant exponent	Inertia subgroup	Galois tame degree	Indices

Residue field degree	Top slope	Wild	Visible	Wild inertia subgroup

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Results (1-50 of 258 matches)

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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)	$5$	$2$	$[3/2]$	$[3/2]_{2}^{5}$	$[4, 0]$	$[1, 1]$

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.