LMFDB - p-adic field search results

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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 59 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
2.4.0.1	$4$	x⁴ + x + 1	$2$	$1$	$4$	$0$	$C_4$ (as 4T1)	$4$	$1$	$[\ ]$	$[\ ]^{4}$	$[0]$	$[]$
2.4.4.1	$4$	x⁴ + 6x³ + 17x² + 24x + 13	$2$	$2$	$2$	$4$	$C_2^2$ (as 4T2)	$2$	$1$	$[2]$	$[2]^{2}$	$[1, 0]$	$[1]$
2.4.4.2	$4$	x⁴ + 4x³ + 4x² + 12	$2$	$2$	$2$	$4$	$C_4$ (as 4T1)	$2$	$1$	$[2]$	$[2]^{2}$	$[1, 0]$	$[1]$
2.4.4.3	$4$	x⁴ + 2x³ + 4x² + 12x + 12	$2$	$2$	$2$	$4$	$D_{4}$ (as 4T3)	$2$	$1$	$[2]$	$[2, 2]^{2}$	$[1, 0]$	$[1]$
2.4.4.4	$4$	x⁴ - 2x³ + 4x² + 12x + 12	$2$	$2$	$2$	$4$	$D_{4}$ (as 4T3)	$2$	$1$	$[2]$	$[2, 2]^{2}$	$[1, 0]$	$[1]$
2.4.4.5	$4$	x⁴ + 2x + 2	$2$	$4$	$1$	$4$	$S_4$ (as 4T5)	$2$	$3$	$[4/3, 4/3]$	$[4/3, 4/3]_{3}^{2}$	$[1, 1, 0]$	$[1]$
2.4.6.1	$4$	x⁴ + 2x³ + 31x² + 30x + 183	$2$	$2$	$2$	$6$	$C_2^2$ (as 4T2)	$2$	$1$	$[3]$	$[3]^{2}$	$[2, 0]$	$[1]$
2.4.6.2	$4$	x⁴ + 4x³ + 16x² + 24x + 12	$2$	$2$	$2$	$6$	$C_2^2$ (as 4T2)	$2$	$1$	$[3]$	$[3]^{2}$	$[2, 0]$	$[1]$
2.4.6.3	$4$	x⁴ + 8x³ + 28x² + 48x + 84	$2$	$2$	$2$	$6$	$C_4$ (as 4T1)	$2$	$1$	$[3]$	$[3]^{2}$	$[2, 0]$	$[1]$
2.4.6.4	$4$	x⁴ + 4x³ + 24x² + 88x + 124	$2$	$2$	$2$	$6$	$C_4$ (as 4T1)	$2$	$1$	$[3]$	$[3]^{2}$	$[2, 0]$	$[1]$
2.4.6.5	$4$	x⁴ - 4x³ + 36x² + 8x + 148	$2$	$2$	$2$	$6$	$D_{4}$ (as 4T3)	$2$	$1$	$[3]$	$[2, 3]^{2}$	$[2, 0]$	$[1]$
2.4.6.6	$4$	x⁴ - 4x³ + 28x² - 24x + 36	$2$	$2$	$2$	$6$	$D_{4}$ (as 4T3)	$2$	$1$	$[3]$	$[2, 3]^{2}$	$[2, 0]$	$[1]$
2.4.6.7	$4$	x⁴ + 2x³ + 2x² + 2	$2$	$4$	$1$	$6$	$A_4$ (as 4T4)	$3$	$1$	$[2, 2]$	$[2, 2]^{3}$	$[3, 2, 0]$	$[3]$
2.4.6.8	$4$	x⁴ + 2x³ + 2	$2$	$4$	$1$	$6$	$D_{4}$ (as 4T3)	$2$	$1$	$[2, 2]$	$[2, 2]^{2}$	$[3, 3, 0]$	$[2]$
2.4.6.9	$4$	x⁴ + 2x³ + 6	$2$	$4$	$1$	$6$	$D_{4}$ (as 4T3)	$2$	$1$	$[2, 2]$	$[2, 2]^{2}$	$[3, 3, 0]$	$[2]$
2.4.8.1	$4$	x⁴ + 2x² + 4x + 10	$2$	$4$	$1$	$8$	$C_2^2$ (as 4T2)	$1$	$1$	$[2, 3]$	$[2, 3]$	$[5, 2, 0]$	$[1, 1]$
2.4.8.2	$4$	x⁴ + 2x² + 4x + 2	$2$	$4$	$1$	$8$	$C_2^2$ (as 4T2)	$1$	$1$	$[2, 3]$	$[2, 3]$	$[5, 2, 0]$	$[1, 1]$
2.4.8.3	$4$	x⁴ + 6x² + 4x + 14	$2$	$4$	$1$	$8$	$C_2^2$ (as 4T2)	$1$	$1$	$[2, 3]$	$[2, 3]$	$[5, 2, 0]$	$[1, 1]$
2.4.8.4	$4$	x⁴ + 6x² + 4x + 6	$2$	$4$	$1$	$8$	$C_2^2$ (as 4T2)	$1$	$1$	$[2, 3]$	$[2, 3]$	$[5, 2, 0]$	$[1, 1]$
2.4.8.5	$4$	x⁴ + 2x² + 4x + 6	$2$	$4$	$1$	$8$	$D_{4}$ (as 4T3)	$2$	$1$	$[2, 3]$	$[2, 3]^{2}$	$[5, 2, 0]$	$[1, 1]$
2.4.8.6	$4$	x⁴ + 6x² + 4x + 10	$2$	$4$	$1$	$8$	$D_{4}$ (as 4T3)	$2$	$1$	$[2, 3]$	$[2, 3]^{2}$	$[5, 2, 0]$	$[1, 1]$
2.4.8.7	$4$	x⁴ + 4x² + 4x + 2	$2$	$4$	$1$	$8$	$S_4$ (as 4T5)	$2$	$3$	$[8/3, 8/3]$	$[8/3, 8/3]_{3}^{2}$	$[5, 4, 0]$	$[1]$
2.4.8.8	$4$	x⁴ + 4x + 2	$2$	$4$	$1$	$8$	$S_4$ (as 4T5)	$2$	$3$	$[8/3, 8/3]$	$[8/3, 8/3]_{3}^{2}$	$[5, 4, 0]$	$[1]$
2.4.9.1	$4$	x⁴ + 10x² + 2	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 7/2]$	$[2, 3, 7/2]$	$[6, 2, 0]$	$[1, 1]$
2.4.9.2	$4$	x⁴ + 2x² + 2	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 7/2]$	$[2, 3, 7/2]$	$[6, 2, 0]$	$[1, 1]$
2.4.9.3	$4$	x⁴ + 4x³ + 2x² + 2	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 7/2]$	$[2, 3, 7/2]$	$[6, 2, 0]$	$[1, 1]$
2.4.9.4	$4$	x⁴ + 10x² + 8x + 2	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 7/2]$	$[2, 3, 7/2]$	$[6, 2, 0]$	$[1, 1]$
2.4.9.5	$4$	x⁴ + 10x² + 8x + 6	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 7/2]$	$[2, 3, 7/2]$	$[6, 2, 0]$	$[1, 1]$
2.4.9.6	$4$	x⁴ + 4x³ + 10x² + 14	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 7/2]$	$[2, 3, 7/2]$	$[6, 2, 0]$	$[1, 1]$
2.4.9.7	$4$	x⁴ + 2x² + 6	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 7/2]$	$[2, 3, 7/2]$	$[6, 2, 0]$	$[1, 1]$
2.4.9.8	$4$	x⁴ + 10x² + 6	$2$	$4$	$1$	$9$	$D_{4}$ (as 4T3)	$1$	$1$	$[2, 7/2]$	$[2, 3, 7/2]$	$[6, 2, 0]$	$[1, 1]$
2.4.10.1	$4$	x⁴ + 4x³ + 10	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 7/2]$	$[2, 3, 7/2]$	$[7, 4, 0]$	$[1, 1]$
2.4.10.2	$4$	x⁴ + 4x³ + 2	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 7/2]$	$[2, 3, 7/2]$	$[7, 4, 0]$	$[1, 1]$
2.4.10.3	$4$	x⁴ + 4x³ + 8x² + 2	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 7/2]$	$[2, 3, 7/2]$	$[7, 4, 0]$	$[1, 1]$
2.4.10.4	$4$	x⁴ + 4x³ + 8x² + 10	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 7/2]$	$[2, 3, 7/2]$	$[7, 4, 0]$	$[1, 1]$
2.4.10.5	$4$	x⁴ + 4x³ + 4x² + 10	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 7/2]$	$[2, 3, 7/2]$	$[7, 4, 0]$	$[1, 1]$
2.4.10.6	$4$	x⁴ + 4x³ + 12x² + 10	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 7/2]$	$[2, 3, 7/2]$	$[7, 4, 0]$	$[1, 1]$
2.4.10.7	$4$	x⁴ + 4x³ + 4x² + 2	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 7/2]$	$[2, 3, 7/2]$	$[7, 4, 0]$	$[1, 1]$
2.4.10.8	$4$	x⁴ + 4x³ + 12x² + 2	$2$	$4$	$1$	$10$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 7/2]$	$[2, 3, 7/2]$	$[7, 4, 0]$	$[1, 1]$
2.4.11.1	$4$	x⁴ + 8x³ + 4x² + 2	$2$	$4$	$1$	$11$	$C_4$ (as 4T1)	$1$	$1$	$[3, 4]$	$[3, 4]$	$[8, 4, 0]$	$[1, 1]$
2.4.11.10	$4$	x⁴ + 4x² + 26	$2$	$4$	$1$	$11$	$C_4$ (as 4T1)	$1$	$1$	$[3, 4]$	$[3, 4]$	$[8, 4, 0]$	$[1, 1]$
2.4.11.11	$4$	x⁴ + 10	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 4]$	$[2, 3, 4]$	$[8, 4, 0]$	$[1, 1]$
2.4.11.12	$4$	x⁴ + 26	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 4]$	$[2, 3, 4]$	$[8, 4, 0]$	$[1, 1]$
2.4.11.13	$4$	x⁴ + 8x³ + 8x + 10	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$2$	$1$	$[3, 4]$	$[3, 4]^{2}$	$[8, 4, 0]$	$[1, 1]$
2.4.11.14	$4$	x⁴ + 8x + 10	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$2$	$1$	$[3, 4]$	$[3, 4]^{2}$	$[8, 4, 0]$	$[1, 1]$
2.4.11.15	$4$	x⁴ + 12x² + 8x + 2	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 4]$	$[2, 3, 4]$	$[8, 4, 0]$	$[1, 1]$
2.4.11.16	$4$	x⁴ + 12x² + 8x + 18	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 4]$	$[2, 3, 4]$	$[8, 4, 0]$	$[1, 1]$
2.4.11.17	$4$	x⁴ + 8x³ + 8x + 2	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$2$	$1$	$[3, 4]$	$[3, 4]^{2}$	$[8, 4, 0]$	$[1, 1]$
2.4.11.18	$4$	x⁴ + 8x + 18	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$2$	$1$	$[3, 4]$	$[3, 4]^{2}$	$[8, 4, 0]$	$[1, 1]$
2.4.11.19	$4$	x⁴ + 12x² + 8x + 26	$2$	$4$	$1$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 4]$	$[2, 3, 4]$	$[8, 4, 0]$	$[1, 1]$

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