$p$-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 59 matches)

 

Label	$n$	Polynomial	$p$	$e$	$f$	$c$	Galois group	$u$	$t$	Slope content
2.4.0.1	$4$	x^4 - x + 1	$2$	$1$	$4$	$0$	$C_4$ (as 4T1)	$4$	$1$	$[\ ]^{4}$
2.4.4.1	$4$	x^4 + 8x^2 + 4	$2$	$2$	$2$	$4$	$C_2^2$ (as 4T2)	$2$	$1$	$[2]^{2}$
2.4.4.2	$4$	x^4 - x^2 + 5	$2$	$2$	$2$	$4$	$C_4$ (as 4T1)	$2$	$1$	$[2]^{2}$
2.4.4.3	$4$	x^4 + 2x^2 + 4x + 4	$2$	$2$	$2$	$4$	$D_{4}$ (as 4T3)	$2$	$1$	$[2, 2]^{2}$
2.4.4.4	$4$	x^4 - 5	$2$	$2$	$2$	$4$	$D_{4}$ (as 4T3)	$2$	$1$	$[2, 2]^{2}$
2.4.4.5	$4$	x^4 + 2x + 2	$2$	$4$	$1$	$4$	$S_4$ (as 4T5)	$2$	$3$	$[4/3, 4/3]_{3}^{2}$
2.4.6.1	$4$	x^4 - 6x^2 + 4	$2$	$2$	$2$	$6$	$C_2^2$ (as 4T2)	$2$	$1$	$[3]^{2}$
2.4.6.2	$4$	x^4 - 2x^2 + 4	$2$	$2$	$2$	$6$	$C_2^2$ (as 4T2)	$2$	$1$	$[3]^{2}$
2.4.6.3	$4$	x^4 + 2x^2 + 20	$2$	$2$	$2$	$6$	$C_4$ (as 4T1)	$2$	$1$	$[3]^{2}$
2.4.6.4	$4$	x^4 - 2x^2 + 20	$2$	$2$	$2$	$6$	$C_4$ (as 4T1)	$2$	$1$	$[3]^{2}$
2.4.6.5	$4$	x^4 + 2x^2 - 4	$2$	$2$	$2$	$6$	$D_{4}$ (as 4T3)	$2$	$1$	$[2, 3]^{2}$
2.4.6.6	$4$	x^4 - 20	$2$	$2$	$2$	$6$	$D_{4}$ (as 4T3)	$2$	$1$	$[2, 3]^{2}$
2.4.6.7	$4$	x^4 + 2x^3 + 2x^2 + 2	$2$	$4$	$1$	$6$	$A_4$ (as 4T4)	$3$	$1$	$[2, 2]^{3}$
2.4.6.8	$4$	x^4 + 2x^3 + 2	$2$	$4$	$1$	$6$	$D_{4}$ (as 4T3)	$2$	$1$	$[2, 2]^{2}$
2.4.6.9	$4$	x^4 + 2x^3 + 6	$2$	$4$	$1$	$6$	$D_{4}$ (as 4T3)	$2$	$1$	$[2, 2]^{2}$
2.4.8.1	$4$	x^4 + 2x^2 + 4x + 10	$2$	$4$	$1$	$8$	$C_2^2$ (as 4T2)	$1$	$1$	$[2, 3]$
2.4.8.2	$4$	x^4 + 6x^2 + 1	$2$	$4$	$1$	$8$	$C_2^2$ (as 4T2)	$1$	$1$	$[2, 3]$
2.4.8.3	$4$	x^4 + 6x^2 + 4x + 14	$2$	$4$	$1$	$8$	$C_2^2$ (as 4T2)	$1$	$1$	$[2, 3]$
2.4.8.4	$4$	x^4 + 6x^2 + 4x + 6	$2$	$4$	$1$	$8$	$C_2^2$ (as 4T2)	$1$	$1$	$[2, 3]$
2.4.8.5	$4$	x^4 + 2x^2 + 4x + 6	$2$	$4$	$1$	$8$	$D_{4}$ (as 4T3)	$2$	$1$	$[2, 3]^{2}$
2.4.8.6	$4$	x^4 + 6x^2 + 4x + 2	$2$	$4$	$1$	$8$	$D_{4}$ (as 4T3)	$2$	$1$	$[2, 3]^{2}$
2.4.8.7	$4$	x^4 + 4x^2 + 4x + 2	$2$	$4$	$1$	$8$	$S_4$ (as 4T5)	$2$	$3$	$[8/3, 8/3]_{3}^{2}$
2.4.8.8	$4$	x^4 + 4x + 2	$2$	$4$	$1$	$8$	$S_4$ (as 4T5)	$2$	$3$	$[8/3, 8/3]_{3}^{2}$
2.4.9.1	$4$	x^4 + 6x^2 + 2	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.9.2	$4$	x^4 - 2x^2 + 2	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.9.3	$4$	x^4 + 6x^2 + 10	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.9.4	$4$	x^4 + 2x^2 + 10	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.9.5	$4$	x^4 + 2x^2 - 2	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.9.6	$4$	x^4 - 2x^2 - 2	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.9.7	$4$	x^4 + 2x^2 + 6	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.9.8	$4$	x^4 - 2x^2 + 6	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.10.1	$4$	x^4 + 2x^2 - 9	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.10.2	$4$	x^4 + 2x^2 - 1	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.10.3	$4$	x^4 + 6x^2 - 9	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.10.4	$4$	x^4 + 6x^2 - 1	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.10.5	$4$	x^4 - 6x^2 + 3	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.10.6	$4$	x^4 + 6x^2 + 3	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.10.7	$4$	x^4 - 2x^2 + 3	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.10.8	$4$	x^4 + 2x^2 + 3	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 7/2]$
2.4.11.1	$4$	x^4 + 12x^2 + 2	$2$	$4$	$1$	$11$	$C_4$ (as 4T1)	$1$	$1$	$[3, 4]$
2.4.11.10	$4$	x^4 + 8x + 6	$2$	$4$	$1$	$11$	$C_4$ (as 4T1)	$1$	$1$	$[3, 4]$
2.4.11.11	$4$	x^4 + 10	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 4]$
2.4.11.12	$4$	x^4 + 26	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 4]$
2.4.11.13	$4$	x^4 + 4x^2 + 14	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$2$	$1$	$[3, 4]^{2}$
2.4.11.14	$4$	x^4 + 8x + 10	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$2$	$1$	$[3, 4]^{2}$
2.4.11.15	$4$	x^4 + 30	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 4]$
2.4.11.16	$4$	x^4 + 14	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 4]$
2.4.11.17	$4$	x^4 + 4x^2 + 6	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$2$	$1$	$[3, 4]^{2}$
2.4.11.18	$4$	x^4 + 12x^2 + 6	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$2$	$1$	$[3, 4]^{2}$
2.4.11.19	$4$	x^4 + 22	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 3, 4]$