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 (26 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
5.5.0.1	$5$	x⁵ + 4x + 3	$5$	$1$	$5$	$0$	$C_5$ (as 5T1)	$5$	$1$	$[\ ]$	$[\ ]^{5}$	$[0]$	$[]$
5.5.5.1	$5$	x⁵ + 20x + 5	$5$	$5$	$1$	$5$	$F_5$ (as 5T3)	$1$	$4$	$[5/4]$	$[5/4]_{4}$	$[1, 0]$	$[1]$
5.5.5.2	$5$	x⁵ + 5x + 5	$5$	$5$	$1$	$5$	$F_5$ (as 5T3)	$1$	$4$	$[5/4]$	$[5/4]_{4}$	$[1, 0]$	$[1]$
5.5.5.3	$5$	x⁵ + 15x + 5	$5$	$5$	$1$	$5$	$F_5$ (as 5T3)	$1$	$4$	$[5/4]$	$[5/4]_{4}$	$[1, 0]$	$[1]$
5.5.5.4	$5$	x⁵ + 10x + 5	$5$	$5$	$1$	$5$	$F_5$ (as 5T3)	$1$	$4$	$[5/4]$	$[5/4]_{4}$	$[1, 0]$	$[1]$
5.5.6.1	$5$	x⁵ + 10x² + 5	$5$	$5$	$1$	$6$	$D_{5}$ (as 5T2)	$1$	$2$	$[3/2]$	$[3/2]_{2}$	$[2, 0]$	$[1]$
5.5.6.2	$5$	x⁵ + 15x² + 5	$5$	$5$	$1$	$6$	$D_{5}$ (as 5T2)	$1$	$2$	$[3/2]$	$[3/2]_{2}$	$[2, 0]$	$[1]$
5.5.6.3	$5$	x⁵ + 5x² + 5	$5$	$5$	$1$	$6$	$F_5$ (as 5T3)	$2$	$2$	$[3/2]$	$[3/2]_{2}^{2}$	$[2, 0]$	$[2]$
5.5.6.4	$5$	x⁵ + 20x² + 5	$5$	$5$	$1$	$6$	$F_5$ (as 5T3)	$2$	$2$	$[3/2]$	$[3/2]_{2}^{2}$	$[2, 0]$	$[2]$
5.5.7.1	$5$	x⁵ + 15x³ + 5	$5$	$5$	$1$	$7$	$F_5$ (as 5T3)	$1$	$4$	$[7/4]$	$[7/4]_{4}$	$[3, 0]$	$[1]$
5.5.7.2	$5$	x⁵ + 10x³ + 5	$5$	$5$	$1$	$7$	$F_5$ (as 5T3)	$1$	$4$	$[7/4]$	$[7/4]_{4}$	$[3, 0]$	$[1]$
5.5.7.3	$5$	x⁵ + 20x³ + 5	$5$	$5$	$1$	$7$	$F_5$ (as 5T3)	$1$	$4$	$[7/4]$	$[7/4]_{4}$	$[3, 0]$	$[1]$
5.5.7.4	$5$	x⁵ + 5x³ + 5	$5$	$5$	$1$	$7$	$F_5$ (as 5T3)	$1$	$4$	$[7/4]$	$[7/4]_{4}$	$[3, 0]$	$[1]$
5.5.8.1	$5$	x⁵ + 20x⁴ + 105	$5$	$5$	$1$	$8$	$C_5$ (as 5T1)	$1$	$1$	$[2]$	$[2]$	$[4, 0]$	$[1]$
5.5.8.2	$5$	x⁵ + 20x⁴ + 5	$5$	$5$	$1$	$8$	$C_5$ (as 5T1)	$1$	$1$	$[2]$	$[2]$	$[4, 0]$	$[1]$
5.5.8.3	$5$	x⁵ + 20x⁴ + 30	$5$	$5$	$1$	$8$	$C_5$ (as 5T1)	$1$	$1$	$[2]$	$[2]$	$[4, 0]$	$[1]$
5.5.8.4	$5$	x⁵ + 20x⁴ + 55	$5$	$5$	$1$	$8$	$C_5$ (as 5T1)	$1$	$1$	$[2]$	$[2]$	$[4, 0]$	$[1]$
5.5.8.5	$5$	x⁵ + 20x⁴ + 80	$5$	$5$	$1$	$8$	$C_5$ (as 5T1)	$1$	$1$	$[2]$	$[2]$	$[4, 0]$	$[1]$
5.5.8.6	$5$	x⁵ + 5x⁴ + 5	$5$	$5$	$1$	$8$	$D_{5}$ (as 5T2)	$2$	$1$	$[2]$	$[2]^{2}$	$[4, 0]$	$[2]$
5.5.8.7	$5$	x⁵ + 10x⁴ + 5	$5$	$5$	$1$	$8$	$F_5$ (as 5T3)	$4$	$1$	$[2]$	$[2]^{4}$	$[4, 0]$	$[4]$
5.5.8.8	$5$	x⁵ + 15x⁴ + 5	$5$	$5$	$1$	$8$	$F_5$ (as 5T3)	$4$	$1$	$[2]$	$[2]^{4}$	$[4, 0]$	$[4]$
5.5.9.1	$5$	x⁵ + 5	$5$	$5$	$1$	$9$	$F_5$ (as 5T3)	$1$	$4$	$[9/4]$	$[9/4]_{4}$	$[5, 0]$	$[1]$
5.5.9.2	$5$	x⁵ + 100x² + 50x + 5	$5$	$5$	$1$	$9$	$F_5$ (as 5T3)	$1$	$4$	$[9/4]$	$[9/4]_{4}$	$[5, 0]$	$[1]$
5.5.9.3	$5$	x⁵ + 100x² + 75x + 5	$5$	$5$	$1$	$9$	$F_5$ (as 5T3)	$1$	$4$	$[9/4]$	$[9/4]_{4}$	$[5, 0]$	$[1]$
5.5.9.4	$5$	x⁵ + 25x² + 25x + 5	$5$	$5$	$1$	$9$	$F_5$ (as 5T3)	$1$	$4$	$[9/4]$	$[9/4]_{4}$	$[5, 0]$	$[1]$
5.5.9.5	$5$	x⁵ + 25x² + 100x + 5	$5$	$5$	$1$	$9$	$F_5$ (as 5T3)	$1$	$4$	$[9/4]$	$[9/4]_{4}$	$[5, 0]$	$[1]$

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