$p$-adic field search results

Degree	Residue characteristic	Galois group	Galois unramified degree
Ramification index	Discriminant exponent	Inertia subgroup	Galois tame degree
Residue field degree	Top slope	Wild inertia subgroup

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Results (1-50 of 258 matches)

 

Label	$n$	Polynomial	$p$	$e$	$f$	$c$	Galois group	$u$	$t$	Slope content
5.10.0.1	$10$	x^10 + x^2 - x + 3	$5$	$1$	$10$	$0$	$C_{10}$ (as 10T1)	$10$	$1$	$[\ ]^{10}$
5.10.5.1	$10$	x^10 - 50x^6 + 625x^2 - 12500	$5$	$2$	$5$	$5$	$C_{10}$ (as 10T1)	$5$	$2$	$[\ ]_{2}^{5}$
5.10.5.2	$10$	x^10 - 625x^2 + 6250	$5$	$2$	$5$	$5$	$C_{10}$ (as 10T1)	$5$	$2$	$[\ ]_{2}^{5}$
5.10.10.1	$10$	x^10 + 10x^5 + 75x^2 + 25	$5$	$5$	$2$	$10$	$C_5^2 : C_8$ (as 10T18)	$2$	$4$	$[5/4, 5/4]_{4}^{2}$
5.10.10.10	$10$	x^10 + 10x^8 + 5x^6 + 10x^5 - 20x^4 - 20x^2 + 2	$5$	$5$	$2$	$10$	$F_{5}\times C_2$ (as 10T5)	$2$	$4$	$[5/4]_{4}^{2}$
5.10.10.11	$10$	x^10 + 5x^6 + 10x^5 + 25x^2 + 25x + 25	$5$	$5$	$2$	$10$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$2$	$4$	$[5/4, 5/4]_{4}^{2}$
5.10.10.12	$10$	x^10 + 20x^6 + 10x^5 + 25x^2 + 100x + 25	$5$	$5$	$2$	$10$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$2$	$4$	$[5/4, 5/4]_{4}^{2}$
5.10.10.13	$10$	x^10 + 15x^6 + 10x^5 + 100x^2 + 75x + 25	$5$	$5$	$2$	$10$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$2$	$4$	$[5/4, 5/4]_{4}^{2}$
5.10.10.14	$10$	x^10 + 10x^6 + 10x^5 + 100x^2 + 50x + 25	$5$	$5$	$2$	$10$	$(C_5^2 : C_4) : C_2$ (as 10T17)	$2$	$4$	$[5/4, 5/4]_{4}^{2}$
5.10.10.15	$10$	x^10 + 20x + 5	$5$	$10$	$1$	$10$	$(C_5^2 : C_8):C_2$ (as 10T28)	$2$	$8$	$[9/8, 9/8]_{8}^{2}$
5.10.10.16	$10$	x^10 + 10x + 5	$5$	$10$	$1$	$10$	$(C_5^2 : C_8):C_2$ (as 10T28)	$2$	$8$	$[9/8, 9/8]_{8}^{2}$
5.10.10.17	$10$	x^10 + 20x + 10	$5$	$10$	$1$	$10$	$(C_5^2 : C_8):C_2$ (as 10T28)	$2$	$8$	$[9/8, 9/8]_{8}^{2}$
5.10.10.18	$10$	x^10 + 15x + 10	$5$	$10$	$1$	$10$	$(C_5^2 : C_8):C_2$ (as 10T28)	$2$	$8$	$[9/8, 9/8]_{8}^{2}$
5.10.10.2	$10$	x^10 + 5x^6 + 10x^5 + 50x^2 + 25x + 25	$5$	$5$	$2$	$10$	$C_5^2 : C_8$ (as 10T18)	$2$	$4$	$[5/4, 5/4]_{4}^{2}$
5.10.10.3	$10$	x^10 + 10x^5 + 50x^2 + 25	$5$	$5$	$2$	$10$	$C_5^2 : C_8$ (as 10T18)	$2$	$4$	$[5/4, 5/4]_{4}^{2}$
5.10.10.4	$10$	x^10 + 20x^6 + 10x^5 + 50x^2 + 100x + 25	$5$	$5$	$2$	$10$	$C_5^2 : C_8$ (as 10T18)	$2$	$4$	$[5/4, 5/4]_{4}^{2}$
5.10.10.5	$10$	x^10 + 15x^6 + 10x^5 + 75x^2 + 75x + 25	$5$	$5$	$2$	$10$	$C_5^2 : C_8$ (as 10T18)	$2$	$4$	$[5/4, 5/4]_{4}^{2}$
5.10.10.6	$10$	x^10 + 10x^6 + 10x^5 + 75x^2 + 50x + 25	$5$	$5$	$2$	$10$	$C_5^2 : C_8$ (as 10T18)	$2$	$4$	$[5/4, 5/4]_{4}^{2}$
5.10.10.7	$10$	x^10 + 10x^8 + 10x^5 - 20x^4 - 20x^2 + 12	$5$	$5$	$2$	$10$	$F_{5}\times C_2$ (as 10T5)	$2$	$4$	$[5/4]_{4}^{2}$
5.10.10.8	$10$	x^10 + 10x^8 + 20x^6 + 10x^5 - 20x^4 - 20x^2 + 22	$5$	$5$	$2$	$10$	$F_{5}\times C_2$ (as 10T5)	$2$	$4$	$[5/4]_{4}^{2}$
5.10.10.9	$10$	x^10 + 10x^8 + 10x^6 + 10x^5 - 20x^4 - 20x^2 + 17	$5$	$5$	$2$	$10$	$F_{5}\times C_2$ (as 10T5)	$2$	$4$	$[5/4]_{4}^{2}$
5.10.11.1	$10$	x^10 + 20x^2 + 5	$5$	$10$	$1$	$11$	$F_5$ (as 10T4)	$1$	$4$	$[5/4]_{4}$
5.10.11.2	$10$	x^10 + 5x^2 + 5	$5$	$10$	$1$	$11$	$F_5$ (as 10T4)	$1$	$4$	$[5/4]_{4}$
5.10.11.3	$10$	x^10 + 10x^2 + 5	$5$	$10$	$1$	$11$	$F_{5}\times C_2$ (as 10T5)	$2$	$4$	$[5/4]_{4}^{2}$
5.10.11.4	$10$	x^10 + 15x^2 + 5	$5$	$10$	$1$	$11$	$F_{5}\times C_2$ (as 10T5)	$2$	$4$	$[5/4]_{4}^{2}$
5.10.11.5	$10$	x^10 + 15x^2 + 10	$5$	$10$	$1$	$11$	$F_5$ (as 10T4)	$1$	$4$	$[5/4]_{4}$
5.10.11.6	$10$	x^10 + 10x^2 + 10	$5$	$10$	$1$	$11$	$F_5$ (as 10T4)	$1$	$4$	$[5/4]_{4}$
5.10.11.7	$10$	x^10 + 5x^2 + 10	$5$	$10$	$1$	$11$	$F_{5}\times C_2$ (as 10T5)	$2$	$4$	$[5/4]_{4}^{2}$
5.10.11.8	$10$	x^10 + 20x^2 + 10	$5$	$10$	$1$	$11$	$F_{5}\times C_2$ (as 10T5)	$2$	$4$	$[5/4]_{4}^{2}$
5.10.12.1	$10$	x^10 + 10x^5 + 75x^4 + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_8$ (as 10T18)	$4$	$2$	$[3/2, 3/2]_{2}^{4}$
5.10.12.10	$10$	x^10 + 10x^8 + 20x^7 + 15x^6 - 5x^5 + 5x^4 + 5x^2 - 5x + 7	$5$	$5$	$2$	$12$	$D_{10}$ (as 10T3)	$2$	$2$	$[3/2]_{2}^{2}$
5.10.12.11	$10$	x^10 + 5x^7 + 10x^5 + 25x^4 + 25x^2 + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_4$ (as 10T10)	$2$	$2$	$[3/2, 3/2]_{2}^{2}$
5.10.12.12	$10$	x^10 + 20x^7 + 10x^5 + 25x^4 + 100x^2 + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_4$ (as 10T10)	$2$	$2$	$[3/2, 3/2]_{2}^{2}$
5.10.12.13	$10$	x^10 + 15x^7 + 10x^5 + 100x^4 + 75x^2 + 25	$5$	$5$	$2$	$12$	$D_5^2$ (as 10T9)	$2$	$2$	$[3/2, 3/2]_{2}^{2}$
5.10.12.14	$10$	x^10 + 10x^7 + 10x^5 + 100x^4 + 50x^2 + 25	$5$	$5$	$2$	$12$	$D_5^2$ (as 10T9)	$2$	$2$	$[3/2, 3/2]_{2}^{2}$
5.10.12.15	$10$	x^10 + 20x^3 + 5	$5$	$10$	$1$	$12$	$(C_5^2 : C_8):C_2$ (as 10T28)	$2$	$8$	$[11/8, 11/8]_{8}^{2}$
5.10.12.16	$10$	x^10 + 15x^3 + 10	$5$	$10$	$1$	$12$	$(C_5^2 : C_8):C_2$ (as 10T28)	$2$	$8$	$[11/8, 11/8]_{8}^{2}$
5.10.12.17	$10$	x^10 + 10x^3 + 5	$5$	$10$	$1$	$12$	$(C_5^2 : C_8):C_2$ (as 10T28)	$2$	$8$	$[11/8, 11/8]_{8}^{2}$
5.10.12.18	$10$	x^10 + 20x^3 + 10	$5$	$10$	$1$	$12$	$(C_5^2 : C_8):C_2$ (as 10T28)	$2$	$8$	$[11/8, 11/8]_{8}^{2}$
5.10.12.2	$10$	x^10 + 15x^7 + 10x^5 + 75x^4 + 75x^2 + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_8$ (as 10T18)	$4$	$2$	$[3/2, 3/2]_{2}^{4}$
5.10.12.3	$10$	x^10 + 5x^7 + 10x^5 + 50x^4 + 25x^2 + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_8$ (as 10T18)	$4$	$2$	$[3/2, 3/2]_{2}^{4}$
5.10.12.4	$10$	x^10 + 10x^5 + 50x^4 + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_8$ (as 10T18)	$4$	$2$	$[3/2, 3/2]_{2}^{4}$
5.10.12.5	$10$	x^10 + 20x^7 + 10x^5 + 50x^4 + 100x^2 + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_8$ (as 10T18)	$4$	$2$	$[3/2, 3/2]_{2}^{4}$
5.10.12.6	$10$	x^10 + 10x^7 + 10x^5 + 75x^4 + 50x^2 + 25	$5$	$5$	$2$	$12$	$C_5^2 : C_8$ (as 10T18)	$4$	$2$	$[3/2, 3/2]_{2}^{4}$
5.10.12.7	$10$	x^10 + 10x^8 + 5x^7 + 15x^6 + 5x^4 + 5x^2 - 20x + 7	$5$	$5$	$2$	$12$	$D_{10}$ (as 10T3)	$2$	$2$	$[3/2]_{2}^{2}$
5.10.12.8	$10$	x^10 + 10x^8 + 15x^7 + 15x^6 - 20x^5 + 5x^4 + 5x^2 - 10x + 7	$5$	$5$	$2$	$12$	$F_5$ (as 10T4)	$2$	$2$	$[3/2]_{2}^{2}$
5.10.12.9	$10$	x^10 + 10x^8 + 10x^7 + 15x^6 - 10x^5 + 5x^4 + 5x^2 - 15x + 7	$5$	$5$	$2$	$12$	$F_5$ (as 10T4)	$2$	$2$	$[3/2]_{2}^{2}$
5.10.13.1	$10$	x^10 + 15x^4 + 5	$5$	$10$	$1$	$13$	$D_5$ (as 10T2)	$1$	$2$	$[3/2]_{2}$
5.10.13.10	$10$	x^10 + 10x^4 + 10	$5$	$10$	$1$	$13$	$F_{5}\times C_2$ (as 10T5)	$4$	$2$	$[3/2]_{2}^{4}$
5.10.13.11	$10$	x^10 - 15x^5 + 5x^4 + 10	$5$	$10$	$1$	$13$	$D_5\times C_5$ (as 10T6)	$5$	$2$	$[3/2]_{2}^{5}$