LMFDB - p-adic field search results

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Residue field degree	Top slope	Wild inertia subgroup

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Label	$n$	Polynomial	$p$	$e$	$f$	$c$	Galois group	$u$	$t$	Slope content
5.15.0.1	$15$	x¹⁵ + x² + 2	$5$	$1$	$15$	$0$	$C_{15}$ (as 15T1)	$15$	$1$	$[\ ]^{15}$
5.15.10.1	$15$	x¹⁵ - 625x³ + 6250	$5$	$3$	$5$	$10$	$S_3 \times C_5$ (as 15T4)	$10$	$3$	$[\ ]_{3}^{10}$
5.15.15.1	$15$	x¹⁵ + 15x¹⁴ + 15x¹³ + 10x¹² + 10x¹¹ + 2x¹⁰ + 20x⁹ + 15x⁸ + 15x⁷ + 7x⁵ + 15x⁴ + 15x³ + 5x + 17	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.10	$15$	x¹⁵ + 5x¹⁴ + 15x¹³ + 5x¹² + 20x¹¹ + 12x¹⁰ + 20x⁹ + 10x⁷ + 5x⁶ + 22x⁵ + 5x⁴ + 10x³ + 5x + 22	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.11	$15$	x¹⁵ + 10x¹⁴ + 20x¹³ + 15x¹² + 10x¹¹ + 22x¹⁰ + 10x⁹ + 20x⁸ + 5x⁷ + 5x⁶ + 17x⁵ + 5x⁴ + 20x³ + 10x² + 7	$5$	$5$	$3$	$15$	$(C_5^2 : C_4):C_3$ (as 15T19)	$3$	$4$	$[5/4, 5/4]_{4}^{3}$
5.15.15.12	$15$	x¹⁵ + 10x¹⁴ + 20x¹³ + 5x¹² + 15x¹¹ + 7x¹⁰ + 20x⁹ + 10x⁸ + 20x⁷ + 5x⁶ + 7x⁵ + 5x⁴ + 20x³ + 10x² + 5x + 22	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.13	$15$	x¹⁵ + 20x¹² + 7x¹⁰ + 5x⁹ + 15x⁸ + 20x⁷ + 20x⁶ + 7x⁵ + 10x⁴ + 20x³ + 20x² + 20x + 22	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.14	$15$	x¹⁵ + 5x¹⁴ + 15x¹³ + 20x¹² + 20x¹¹ + 2x¹⁰ + 15x⁹ + 5x⁷ + 20x⁶ + 12x⁵ + 20x² + 7	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.15	$15$	x¹⁵ + 10x¹⁴ + 20x¹² + 20x¹¹ + 7x¹⁰ + 5x⁹ + 15x⁸ + 10x⁷ + 10x⁶ + 7x⁵ + 10x⁴ + 20x³ + 15x² + 5x + 17	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.16	$15$	x¹⁵ + 15x¹⁴ + 15x¹³ + 10x¹² + 10x¹¹ + 7x¹⁰ + 15x⁸ + 20x⁷ + 15x⁶ + 17x⁵ + 20x⁴ + 20x³ + 5x² + 5x + 7	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.17	$15$	x¹⁵ + 5x¹⁴ + 20x¹³ + 15x¹¹ + 2x¹⁰ + 5x⁹ + 5x⁶ + 2x⁵ + 15x² + 5x + 17	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.18	$15$	x¹⁵ + 20x¹³ + 10x¹² + 15x¹¹ + 17x¹⁰ + 10x⁹ + 15x⁸ + 20x⁷ + 20x⁶ + 7x⁵ + 20x⁴ + 20x³ + 10x² + 10x + 2	$5$	$5$	$3$	$15$	$F_5\times C_3$ (as 15T8)	$3$	$4$	$[5/4]_{4}^{3}$
5.15.15.19	$15$	x¹⁵ + 5x¹⁴ + 20x¹² + 12x¹⁰ + 5x⁹ + 10x⁸ + 15x⁷ + 12x⁵ + 20x + 22	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.2	$15$	x¹⁵ + 5x¹⁴ + 20x¹³ + 5x¹¹ + 17x¹⁰ + 15x⁹ + 5x⁸ + 20x⁷ + 10x⁶ + 12x⁵ + 15x⁴ + 20x³ + 15x² + 22	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.20	$15$	x¹⁵ + 20x¹³ + 20x¹² + 17x¹⁰ + 20x⁹ + 20x⁸ + 10x⁷ + 5x⁶ + 7x⁵ + 5x⁴ + 15x³ + 20x² + 20x + 12	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.21	$15$	x¹⁵ + 10x¹⁴ + 5x¹³ + 20x¹² + 22x¹⁰ + 15x⁶ + 17x⁵ + 5x³ + 15x² + 2	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.22	$15$	x¹⁵ + 15x¹³ + 5x¹² + 15x¹¹ + 12x¹⁰ + 20x⁹ + 15x⁸ + 15x⁶ + 22x⁵ + 20x⁴ + 10x³ + 2	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.23	$15$	x¹⁵ + 20x¹⁴ + 5x¹³ + 5x¹² + 22x¹⁰ + 15x⁹ + 5x⁸ + 15x⁷ + 10x⁶ + 7x⁵ + 15x² + 15x + 7	$5$	$5$	$3$	$15$	$(C_5^2 : C_4):C_3$ (as 15T19)	$3$	$4$	$[5/4, 5/4]_{4}^{3}$
5.15.15.24	$15$	x¹⁵ + 10x¹⁴ + 20x¹² + 15x¹¹ + 7x¹⁰ + 10x⁹ + 20x⁸ + 5x⁷ + 5x⁶ + 2x⁵ + 10x⁴ + 5x² + 10x + 22	$5$	$5$	$3$	$15$	$(C_5^2 : C_4):C_3$ (as 15T19)	$3$	$4$	$[5/4, 5/4]_{4}^{3}$
5.15.15.25	$15$	x¹⁵ + 5x¹⁴ + 5x¹³ + 20x¹² + 20x¹¹ + 22x¹⁰ + 20x⁸ + 5x⁷ + 12x⁵ + 15x⁴ + 5x² + 20x + 12	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.26	$15$	x¹⁵ + 10x¹⁴ + 10x¹² + 15x¹¹ + 22x¹⁰ + 5x⁹ + 20x⁸ + 5x⁷ + 22x⁵ + 5x⁴ + 15x³ + 5x + 17	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.27	$15$	x¹⁵ + 15x¹⁴ + 5x¹³ + 5x¹² + 7x¹⁰ + 5x⁸ + 5x⁶ + 17x⁵ + 10x⁴ + 20x³ + 20x² + 5x + 2	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.28	$15$	x¹⁵ + 15x¹⁴ + 15x¹³ + 10x¹¹ + 7x¹⁰ + 5x⁸ + 15x⁷ + 15x⁶ + 7x⁵ + 15x⁴ + 15x² + 2	$5$	$5$	$3$	$15$	$(C_5^2 : C_4):C_3$ (as 15T19)	$3$	$4$	$[5/4, 5/4]_{4}^{3}$
5.15.15.29	$15$	x¹⁵ + 20x¹⁴ + 15x¹³ + 12x¹⁰ + 15x⁹ + 10x⁸ + 5x⁷ + 20x⁶ + 2x⁵ + 15x² + 10x + 22	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.3	$15$	x¹⁵ + 10x¹⁴ + 20x¹³ + 2x¹⁰ + 20x⁹ + 20x⁸ + 15x⁷ + 2x⁵ + 10x⁴ + 15x³ + 10x + 7	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.30	$15$	x¹⁵ + 10x¹⁴ + 15x¹³ + 5x¹² + 10x¹¹ + 22x¹⁰ + 5x⁹ + 10x⁸ + 10x⁷ + 5x⁶ + 12x⁵ + 15x⁴ + 10x³ + 5x² + 15x + 7	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.31	$15$	x¹⁵ + 10x¹⁴ + 5x¹³ + 15x¹² + 15x¹¹ + 2x¹⁰ + 15x⁹ + 10x⁷ + 20x⁶ + 17x⁵ + 15x⁴ + 5x³ + 12	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.32	$15$	x¹⁵ + 15x¹⁴ + 5x¹³ + 20x¹² + 5x¹¹ + 12x¹⁰ + 20x⁹ + 15x⁸ + 15x⁶ + 2x⁵ + 10x⁴ + 10x³ + 10x + 12	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.33	$15$	x¹⁵ + 15x¹⁴ + 15x¹³ + 5x¹² + 17x¹⁰ + 5x⁹ + 10x⁷ + 22x⁵ + 5x⁴ + 10x³ + 5x + 7	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.34	$15$	x¹⁵ + 15x¹² + 15x¹¹ + 12x¹⁰ + 20x⁸ + 22x⁵ + 10x⁴ + 20x³ + 15x + 17	$5$	$5$	$3$	$15$	$(C_5^2 : C_4):C_3$ (as 15T19)	$3$	$4$	$[5/4, 5/4]_{4}^{3}$
5.15.15.35	$15$	x¹⁵ + 5x¹⁴ + 15x¹² + 15x¹¹ + 2x¹⁰ + 5x⁹ + 20x⁸ + 2x⁵ + 5x⁴ + 5x³ + 20x² + 20x + 22	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.36	$15$	x¹⁵ + 15x¹⁴ + 5x¹³ + 20x¹² + 15x¹¹ + 12x¹⁰ + 20x⁹ + 15x⁸ + 5x⁷ + 20x⁶ + 12x⁵ + 15x⁴ + 15x² + 20x + 2	$5$	$5$	$3$	$15$	$(C_5^2 : C_4):C_3$ (as 15T19)	$3$	$4$	$[5/4, 5/4]_{4}^{3}$
5.15.15.37	$15$	x¹⁵ + 5x¹⁴ + 10x¹³ + 15x¹² + 20x¹¹ + 12x¹⁰ + 15x⁹ + 15x⁸ + 5x⁷ + 17x⁵ + 15x⁴ + 15x³ + 15x + 2	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.38	$15$	x¹⁵ + 10x¹⁴ + 5x¹³ + 10x¹² + 15x¹¹ + 17x¹⁰ + 20x⁹ + 15x⁸ + 10x⁷ + 17x⁵ + 20x⁴ + 5x³ + 20x + 17	$5$	$5$	$3$	$15$	$F_5\times C_3$ (as 15T8)	$3$	$4$	$[5/4]_{4}^{3}$
5.15.15.39	$15$	x¹⁵ + 10x¹⁴ + 15x¹² + 20x¹¹ + 2x¹⁰ + 5x⁹ + 10x⁸ + 15x⁷ + 15x⁶ + 12x⁵ + 20x⁴ + 10x³ + 10x² + 20x + 12	$5$	$5$	$3$	$15$	$(C_5^2 : C_4):C_3$ (as 15T19)	$3$	$4$	$[5/4, 5/4]_{4}^{3}$
5.15.15.4	$15$	x¹⁵ + 5x¹⁴ + 5x¹³ + 10x¹² + 10x¹¹ + 22x¹⁰ + 10x⁹ + 20x⁷ + 12x⁵ + 20x⁴ + 5x³ + 10x² + 10x + 12	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.40	$15$	x¹⁵ + 10x¹⁴ + 5x¹³ + 20x¹² + 20x¹¹ + 17x¹⁰ + 10x⁹ + 15x⁸ + 12x⁵ + 15x³ + 10x² + 20x + 17	$5$	$5$	$3$	$15$	$F_5\times C_3$ (as 15T8)	$3$	$4$	$[5/4]_{4}^{3}$
5.15.15.41	$15$	x¹⁵ + 10x¹⁴ + 10x¹³ + 15x¹² + 10x¹¹ + 7x¹⁰ + 20x⁹ + 5x⁸ + 15x⁷ + 10x⁶ + 12x⁵ + 10x⁴ + 15x³ + 10x² + 5x + 17	$5$	$5$	$3$	$15$	$(C_5^2 : C_4):C_3$ (as 15T19)	$3$	$4$	$[5/4, 5/4]_{4}^{3}$
5.15.15.42	$15$	x¹⁵ + 15x¹⁴ + 15x¹³ + 20x¹² + 5x¹¹ + 2x¹⁰ + 20x⁸ + 10x⁷ + 10x⁶ + 2x⁵ + 15x³ + 15x² + 15x + 17	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.43	$15$	x¹⁵ + 20x¹⁴ + 15x¹³ + 15x¹² + 5x¹¹ + 12x¹⁰ + 10x⁹ + 20x⁸ + 5x⁷ + 20x⁶ + 22x⁵ + 20x⁴ + 15x² + 5x + 17	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.44	$15$	x¹⁵ + 10x¹⁴ + 15x¹³ + 5x¹² + 20x¹¹ + 17x¹⁰ + 10x⁹ + 5x⁷ + 5x⁶ + 2x⁵ + 10x⁴ + 10x² + 20x + 2	$5$	$5$	$3$	$15$	$F_5\times C_3$ (as 15T8)	$3$	$4$	$[5/4]_{4}^{3}$
5.15.15.45	$15$	x¹⁵ - 5x¹⁴ - 5x¹³ + 5x¹² - 10x¹¹ + 5x¹⁰ - 5x⁹ - 5x⁸ + 5x⁷ + 5x⁶ - 5x⁵ - 10x⁴ + 10x³ + 10x² - 10x + 5	$5$	$15$	$1$	$15$	$C_5^2:(C_4\times S_3)$ (as 15T27)	$2$	$12$	$[13/12, 13/12]_{12}^{2}$
5.15.15.46	$15$	x¹⁵ + 5x¹⁴ - 10x¹² + 10x¹¹ + 10x¹⁰ - 10x⁹ - 10x⁸ - 10x⁷ - 10x⁵ + 5x⁴ - 10x³ - 5x² - 10x - 10	$5$	$15$	$1$	$15$	$C_5^2:(C_4\times S_3)$ (as 15T27)	$2$	$12$	$[13/12, 13/12]_{12}^{2}$
5.15.15.47	$15$	x¹⁵ - 10x¹⁴ - 10x¹³ - 5x¹² - 10x¹¹ - 5x¹⁰ + 10x⁸ + 5x⁶ + 10x⁵ + 10x⁴ - 5x³ + 10x² + 5x - 5	$5$	$15$	$1$	$15$	$C_5^2:(C_4\times S_3)$ (as 15T27)	$2$	$12$	$[13/12, 13/12]_{12}^{2}$
5.15.15.48	$15$	x¹⁵ + 5x¹⁴ - 10x¹³ - 10x¹² + 5x¹¹ - 5x¹⁰ - 10x⁹ + 10x⁸ + 5x⁷ + 5x⁶ - 5x⁵ + 5x⁴ + 5x³ + 10x² - 5x + 5	$5$	$15$	$1$	$15$	$C_5^2:(C_4\times S_3)$ (as 15T27)	$2$	$12$	$[13/12, 13/12]_{12}^{2}$
5.15.15.5	$15$	x¹⁵ + 5x¹⁴ + 15x¹² + 10x¹¹ + 2x¹⁰ + 10x⁹ + 20x⁸ + 20x⁷ + 5x⁶ + 22x⁵ + 10x⁴ + 10x² + 2	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.6	$15$	x¹⁵ + 20x¹⁴ + 5x¹³ + 10x¹² + 15x¹¹ + 2x¹⁰ + 10x⁹ + 5x⁸ + 10x⁷ + 15x⁶ + 17x⁵ + 5x⁴ + 10x³ + 15x² + 15x + 17	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.7	$15$	x¹⁵ + 20x¹⁴ + 5x¹³ + 15x¹² + 7x¹⁰ + 10x⁹ + 20x⁸ + 20x⁷ + 5x⁶ + 7x⁵ + 10x⁴ + 5x³ + 20x + 22	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.8	$15$	x¹⁵ + 15x¹⁴ + 5x¹³ + 20x¹¹ + 7x¹⁰ + 20x⁹ + 5x⁸ + 5x⁶ + 17x⁵ + 15x⁴ + 10x³ + 15x² + 5x + 17	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$
5.15.15.9	$15$	x¹⁵ + 20x¹⁴ + 5x¹³ + 15x¹² + 20x¹¹ + 17x¹⁰ + 15x⁹ + 10x⁶ + 7x⁵ + 15x⁴ + 5x³ + 5x + 12	$5$	$5$	$3$	$15$	$C_5^3:C_{12}$ (as 15T38)	$3$	$4$	$[5/4, 5/4, 5/4]_{4}^{3}$

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