LMFDB - p-adic field search results

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Label	Polynomial	$p$	$e$	$f$	$c$	Galois group	Slope content
5.10.0.1	x¹⁰ + x² - x + 3	$5$	$1$	$10$	$0$	$C_{10}$ (as 10T1)	$ [\ ]^{10}$
5.10.5.1	x¹⁰ - 50x⁶ + 625x² - 12500	$5$	$2$	$5$	$5$	$C_{10}$ (as 10T1)	$ [\ ]_{2}^{5}$
5.10.5.2	x¹⁰ - 625x² + 6250	$5$	$2$	$5$	$5$	$C_{10}$ (as 10T1)	$ [\ ]_{2}^{5}$
5.10.10.1	x¹⁰ + 10x⁵ + 75x² + 25	$5$	$5$	$2$	$10$	$C_5^2 : C_8$ (as 10T18)	$ [5/4, 5/4]_{4}^{2}$
5.10.10.10	x¹⁰ + 10x⁸ + 5x⁶ + 10x⁵ - 20x⁴ - 20x² + 2	$5$	$5$	$2$	$10$	$F_{5}\times C_2$ (as 10T5)	$ [5/4]_{4}^{2}$
5.10.10.11	x¹⁰ + 5x⁶ + 10x⁵ + 25x² + 25x + 25	$5$	$5$	$2$	$10$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$ [5/4, 5/4]_{4}^{2}$
5.10.10.12	x¹⁰ + 20x⁶ + 10x⁵ + 25x² + 100x + 25	$5$	$5$	$2$	$10$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$ [5/4, 5/4]_{4}^{2}$
5.10.10.13	x¹⁰ + 15x⁶ + 10x⁵ + 100x² + 75x + 25	$5$	$5$	$2$	$10$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$ [5/4, 5/4]_{4}^{2}$
5.10.10.14	x¹⁰ + 10x⁶ + 10x⁵ + 100x² + 50x + 25	$5$	$5$	$2$	$10$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$ [5/4, 5/4]_{4}^{2}$
5.10.10.15	x¹⁰ + 20x + 5	$5$	$10$	$1$	$10$	$(C_5^2 : C_8):C_2$ (as 10T28)	$ [9/8, 9/8]_{8}^{2}$
5.10.10.16	x¹⁰ + 10x + 5	$5$	$10$	$1$	$10$	$(C_5^2 : C_8):C_2$ (as 10T28)	$ [9/8, 9/8]_{8}^{2}$
5.10.10.17	x¹⁰ + 20x + 10	$5$	$10$	$1$	$10$	$(C_5^2 : C_8):C_2$ (as 10T28)	$ [9/8, 9/8]_{8}^{2}$
5.10.10.18	x¹⁰ + 15x + 10	$5$	$10$	$1$	$10$	$(C_5^2 : C_8):C_2$ (as 10T28)	$ [9/8, 9/8]_{8}^{2}$
5.10.10.2	x¹⁰ + 5x⁶ + 10x⁵ + 50x² + 25x + 25	$5$	$5$	$2$	$10$	$C_5^2 : C_8$ (as 10T18)	$ [5/4, 5/4]_{4}^{2}$
5.10.10.3	x¹⁰ + 10x⁵ + 50x² + 25	$5$	$5$	$2$	$10$	$C_5^2 : C_8$ (as 10T18)	$ [5/4, 5/4]_{4}^{2}$
5.10.10.4	x¹⁰ + 20x⁶ + 10x⁵ + 50x² + 100x + 25	$5$	$5$	$2$	$10$	$C_5^2 : C_8$ (as 10T18)	$ [5/4, 5/4]_{4}^{2}$
5.10.10.5	x¹⁰ + 15x⁶ + 10x⁵ + 75x² + 75x + 25	$5$	$5$	$2$	$10$	$C_5^2 : C_8$ (as 10T18)	$ [5/4, 5/4]_{4}^{2}$
5.10.10.6	x¹⁰ + 10x⁶ + 10x⁵ + 75x² + 50x + 25	$5$	$5$	$2$	$10$	$C_5^2 : C_8$ (as 10T18)	$ [5/4, 5/4]_{4}^{2}$
5.10.10.7	x¹⁰ + 10x⁸ + 10x⁵ - 20x⁴ - 20x² + 12	$5$	$5$	$2$	$10$	$F_{5}\times C_2$ (as 10T5)	$ [5/4]_{4}^{2}$
5.10.10.8	x¹⁰ + 10x⁸ + 20x⁶ + 10x⁵ - 20x⁴ - 20x² + 22	$5$	$5$	$2$	$10$	$F_{5}\times C_2$ (as 10T5)	$ [5/4]_{4}^{2}$
5.10.10.9	x¹⁰ + 10x⁸ + 10x⁶ + 10x⁵ - 20x⁴ - 20x² + 17	$5$	$5$	$2$	$10$	$F_{5}\times C_2$ (as 10T5)	$ [5/4]_{4}^{2}$
5.10.11.1	x¹⁰ + 20x² + 5	$5$	$10$	$1$	$11$	$F_5$ (as 10T4)	$ [5/4]_{4}$
5.10.11.2	x¹⁰ + 5x² + 5	$5$	$10$	$1$	$11$	$F_5$ (as 10T4)	$ [5/4]_{4}$
5.10.11.3	x¹⁰ + 10x² + 5	$5$	$10$	$1$	$11$	$F_{5}\times C_2$ (as 10T5)	$ [5/4]_{4}^{2}$
5.10.11.4	x¹⁰ + 15x² + 5	$5$	$10$	$1$	$11$	$F_{5}\times C_2$ (as 10T5)	$ [5/4]_{4}^{2}$
5.10.11.5	x¹⁰ + 15x² + 10	$5$	$10$	$1$	$11$	$F_5$ (as 10T4)	$ [5/4]_{4}$
5.10.11.6	x¹⁰ + 10x² + 10	$5$	$10$	$1$	$11$	$F_5$ (as 10T4)	$ [5/4]_{4}$
5.10.11.7	x¹⁰ + 5x² + 10	$5$	$10$	$1$	$11$	$F_{5}\times C_2$ (as 10T5)	$ [5/4]_{4}^{2}$
5.10.11.8	x¹⁰ + 20x² + 10	$5$	$10$	$1$	$11$	$F_{5}\times C_2$ (as 10T5)	$ [5/4]_{4}^{2}$
5.10.12.1	x¹⁰ + 10x⁵ + 75x⁴ + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_8$ (as 10T18)	$ [3/2, 3/2]_{2}^{4}$
5.10.12.10	x¹⁰ + 10x⁸ + 20x⁷ + 15x⁶ - 5x⁵ + 5x⁴ + 5x² - 5x + 7	$5$	$5$	$2$	$12$	$D_{10}$ (as 10T3)	$ [3/2]_{2}^{2}$
5.10.12.11	x¹⁰ + 5x⁷ + 10x⁵ + 25x⁴ + 25x² + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_4$ (as 10T10)	$ [3/2, 3/2]_{2}^{2}$
5.10.12.12	x¹⁰ + 20x⁷ + 10x⁵ + 25x⁴ + 100x² + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_4$ (as 10T10)	$ [3/2, 3/2]_{2}^{2}$
5.10.12.13	x¹⁰ + 15x⁷ + 10x⁵ + 100x⁴ + 75x² + 25	$5$	$5$	$2$	$12$	$D_5^2$ (as 10T9)	$ [3/2, 3/2]_{2}^{2}$
5.10.12.14	x¹⁰ + 10x⁷ + 10x⁵ + 100x⁴ + 50x² + 25	$5$	$5$	$2$	$12$	$D_5^2$ (as 10T9)	$ [3/2, 3/2]_{2}^{2}$
5.10.12.15	x¹⁰ + 20x³ + 5	$5$	$10$	$1$	$12$	$(C_5^2 : C_8):C_2$ (as 10T28)	$ [11/8, 11/8]_{8}^{2}$
5.10.12.16	x¹⁰ + 15x³ + 10	$5$	$10$	$1$	$12$	$(C_5^2 : C_8):C_2$ (as 10T28)	$ [11/8, 11/8]_{8}^{2}$
5.10.12.17	x¹⁰ + 10x³ + 5	$5$	$10$	$1$	$12$	$(C_5^2 : C_8):C_2$ (as 10T28)	$ [11/8, 11/8]_{8}^{2}$
5.10.12.18	x¹⁰ + 20x³ + 10	$5$	$10$	$1$	$12$	$(C_5^2 : C_8):C_2$ (as 10T28)	$ [11/8, 11/8]_{8}^{2}$
5.10.12.2	x¹⁰ + 15x⁷ + 10x⁵ + 75x⁴ + 75x² + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_8$ (as 10T18)	$ [3/2, 3/2]_{2}^{4}$
5.10.12.3	x¹⁰ + 5x⁷ + 10x⁵ + 50x⁴ + 25x² + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_8$ (as 10T18)	$ [3/2, 3/2]_{2}^{4}$
5.10.12.4	x¹⁰ + 10x⁵ + 50x⁴ + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_8$ (as 10T18)	$ [3/2, 3/2]_{2}^{4}$
5.10.12.5	x¹⁰ + 20x⁷ + 10x⁵ + 50x⁴ + 100x² + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_8$ (as 10T18)	$ [3/2, 3/2]_{2}^{4}$
5.10.12.6	x¹⁰ + 10x⁷ + 10x⁵ + 75x⁴ + 50x² + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_8$ (as 10T18)	$ [3/2, 3/2]_{2}^{4}$
5.10.12.7	x¹⁰ + 10x⁸ + 5x⁷ + 15x⁶ + 5x⁴ + 5x² - 20x + 7	$5$	$5$	$2$	$12$	$D_{10}$ (as 10T3)	$ [3/2]_{2}^{2}$
5.10.12.8	x¹⁰ + 10x⁸ + 15x⁷ + 15x⁶ - 20x⁵ + 5x⁴ + 5x² - 10x + 7	$5$	$5$	$2$	$12$	$F_5$ (as 10T4)	$ [3/2]_{2}^{2}$
5.10.12.9	x¹⁰ + 10x⁸ + 10x⁷ + 15x⁶ - 10x⁵ + 5x⁴ + 5x² - 15x + 7	$5$	$5$	$2$	$12$	$F_5$ (as 10T4)	$ [3/2]_{2}^{2}$
5.10.13.1	x¹⁰ + 15x⁴ + 5	$5$	$10$	$1$	$13$	$D_5$ (as 10T2)	$ [3/2]_{2}$
5.10.13.10	x¹⁰ + 10x⁴ + 10	$5$	$10$	$1$	$13$	$F_{5}\times C_2$ (as 10T5)	$ [3/2]_{2}^{4}$
5.10.13.11	x¹⁰ - 15x⁵ + 5x⁴ + 10	$5$	$10$	$1$	$13$	$D_5\times C_5$ (as 10T6)	$ [3/2]_{2}^{5}$

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