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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 (47 matches)

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Label	$n$	Polynomial	$p$	$e$	$f$	$c$	Galois group	$u$	$t$	Visible slopes	Slope content	Unram. Ext.	Eisen. Poly.	Ind. of Insep.	Assoc. Inertia
2.6.0.1	$6$	$x^{6} + x^{4} + x^{3} + x + 1$	$2$	$1$	$6$	$0$	$C_6$ (as 6T1)	$6$	$1$	$[\ ]$	$[\ ]^{6}$	$t^{6} + t^{4} + t^{3} + t + 1$	$x - 2$	$[0]$	$[]$
2.6.4.1	$6$	$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$	$2$	$3$	$2$	$4$	$S_3$ (as 6T2)	$2$	$3$	$[\ ]$	$[\ ]_{3}^{2}$	$t^{2} + t + 1$	$x^{3} + 2 x + 2$	$[0]$	$[1]$
2.6.4.2	$6$	$x^{6} - 2 x^{3} + 4$	$2$	$3$	$2$	$4$	$S_3\times C_3$ (as 6T5)	$6$	$3$	$[\ ]$	$[\ ]_{3}^{6}$	$t^{2} + t + 1$	$x^{3} + 2 t$	$[0]$	$[1]$
2.6.6.1	$6$	$x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$	$2$	$2$	$3$	$6$	$A_4$ (as 6T4)	$3$	$1$	$[2]$	$[2, 2]^{3}$	$t^{3} + t + 1$	$x^{2} + \left(2 t + 2\right) x + 6$	$[1, 0]$	$[1]$
2.6.6.2	$6$	$x^{6} + 6 x^{5} + 14 x^{4} + 24 x^{3} + 44 x^{2} + 8 x + 72$	$2$	$2$	$3$	$6$	$A_4\times C_2$ (as 6T6)	$6$	$1$	$[2]$	$[2, 2]^{6}$	$t^{3} + t + 1$	$x^{2} + \left(2 t + 2\right) x + 4 t^{2} + 2$	$[1, 0]$	$[1]$
2.6.6.3	$6$	$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$	$2$	$2$	$3$	$6$	$C_6$ (as 6T1)	$3$	$1$	$[2]$	$[2]^{3}$	$t^{3} + t + 1$	$x^{2} + 2 x + 2$	$[1, 0]$	$[1]$
2.6.6.4	$6$	$x^{6} - 4 x^{5} + 14 x^{4} - 24 x^{3} + 100 x^{2} + 48 x + 88$	$2$	$2$	$3$	$6$	$A_4\times C_2$ (as 6T6)	$3$	$1$	$[2]$	$[2, 2, 2]^{3}$	$t^{3} + t + 1$	$x^{2} + 2 t^{2} x + 4 t^{2} + 6$	$[1, 0]$	$[1]$
2.6.6.5	$6$	$x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$	$2$	$2$	$3$	$6$	$C_6$ (as 6T1)	$3$	$1$	$[2]$	$[2]^{3}$	$t^{3} + t + 1$	$x^{2} + 2 x + 6$	$[1, 0]$	$[1]$
2.6.6.6	$6$	$x^{6} - 4 x^{5} + 30 x^{4} - 16 x^{3} + 164 x^{2} + 160 x + 88$	$2$	$2$	$3$	$6$	$A_4\times C_2$ (as 6T6)	$3$	$1$	$[2]$	$[2, 2, 2]^{3}$	$t^{3} + t + 1$	$x^{2} + \left(2 t^{2} + 2 t\right) x + 4 t^{2} + 6$	$[1, 0]$	$[1]$
2.6.6.7	$6$	$x^{6} + 2 x^{2} + 2 x + 2$	$2$	$6$	$1$	$6$	$S_4$ (as 6T7)	$2$	$3$	$[4/3]$	$[4/3, 4/3]_{3}^{2}$	$t + 1$	$x^{6} + 2 x^{2} + 2 x + 2$	$[1, 0]$	$[1, 2]$
2.6.6.8	$6$	$x^{6} + 2 x + 2$	$2$	$6$	$1$	$6$	$S_4$ (as 6T8)	$2$	$3$	$[4/3]$	$[4/3, 4/3]_{3}^{2}$	$t + 1$	$x^{6} + 2 x + 2$	$[1, 0]$	$[1, 2]$
2.6.8.1	$6$	$x^{6} + 2 x^{3} + 2$	$2$	$6$	$1$	$8$	$D_{6}$ (as 6T3)	$2$	$3$	$[2]$	$[2]_{3}^{2}$	$t + 1$	$x^{6} + 2 x^{3} + 2$	$[3, 0]$	$[1, 2]$
2.6.8.2	$6$	$x^{6} + 2 x^{4} + 2 x^{3} + 2$	$2$	$6$	$1$	$8$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[2]$	$[4/3, 4/3, 2]_{3}^{2}$	$t + 1$	$x^{6} + 2 x^{4} + 2 x^{3} + 2$	$[3, 0]$	$[1, 2]$
2.6.8.3	$6$	$x^{6} + 2 x^{3} + 6$	$2$	$6$	$1$	$8$	$D_{6}$ (as 6T3)	$2$	$3$	$[2]$	$[2]_{3}^{2}$	$t + 1$	$x^{6} + 2 x^{3} + 6$	$[3, 0]$	$[1, 2]$
2.6.8.4	$6$	$x^{6} + 2 x^{4} + 2 x^{3} + 6$	$2$	$6$	$1$	$8$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[2]$	$[4/3, 4/3, 2]_{3}^{2}$	$t + 1$	$x^{6} + 2 x^{4} + 2 x^{3} + 6$	$[3, 0]$	$[1, 2]$
2.6.9.1	$6$	$x^{6} + 44 x^{4} + 2 x^{3} + 589 x^{2} - 82 x + 2367$	$2$	$2$	$3$	$9$	$C_6$ (as 6T1)	$3$	$1$	$[3]$	$[3]^{3}$	$t^{3} + t + 1$	$x^{2} + 14$	$[2, 0]$	$[1]$
2.6.9.2	$6$	$x^{6} + 4 x^{5} - 10 x^{4} + 160 x^{3} + 1212 x^{2} + 2160 x - 1048$	$2$	$2$	$3$	$9$	$A_4\times C_2$ (as 6T6)	$3$	$1$	$[3]$	$[2, 2, 3]^{3}$	$t^{3} + t + 1$	$x^{2} + \left(4 t^{2} + 4\right) x + 8 t^{2} + 12 t + 2$	$[2, 0]$	$[1]$
2.6.9.3	$6$	$x^{6} + 12 x^{5} + 86 x^{4} + 352 x^{3} + 892 x^{2} + 1552 x + 1384$	$2$	$2$	$3$	$9$	$C_6$ (as 6T1)	$3$	$1$	$[3]$	$[3]^{3}$	$t^{3} + t + 1$	$x^{2} + \left(4 t + 4\right) x + 4 t^{2} + 8 t + 10$	$[2, 0]$	$[1]$
2.6.9.4	$6$	$x^{6} + 4 x^{5} + 14 x^{4} + 224 x^{3} + 1244 x^{2} + 3184 x + 2376$	$2$	$2$	$3$	$9$	$A_4\times C_2$ (as 6T6)	$3$	$1$	$[3]$	$[2, 2, 3]^{3}$	$t^{3} + t + 1$	$x^{2} + \left(4 t^{2} + 4\right) x + 8 t^{2} + 12 t + 10$	$[2, 0]$	$[1]$
2.6.9.5	$6$	$x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243$	$2$	$2$	$3$	$9$	$C_6$ (as 6T1)	$3$	$1$	$[3]$	$[3]^{3}$	$t^{3} + t + 1$	$x^{2} + 4 x + 6$	$[2, 0]$	$[1]$
2.6.9.6	$6$	$x^{6} - 8 x^{5} + 86 x^{4} + 96 x^{3} + 860 x^{2} + 2976 x + 2728$	$2$	$2$	$3$	$9$	$A_4\times C_2$ (as 6T6)	$3$	$1$	$[3]$	$[2, 2, 3]^{3}$	$t^{3} + t + 1$	$x^{2} + \left(4 t^{2} + 4 t\right) x + 12 t^{2} + 4 t + 10$	$[2, 0]$	$[1]$
2.6.9.7	$6$	$x^{6} + 32 x^{4} + 2 x^{3} + 301 x^{2} - 58 x + 811$	$2$	$2$	$3$	$9$	$C_6$ (as 6T1)	$3$	$1$	$[3]$	$[3]^{3}$	$t^{3} + t + 1$	$x^{2} + 10$	$[2, 0]$	$[1]$
2.6.9.8	$6$	$x^{6} - 8 x^{5} + 62 x^{4} + 224 x^{3} + 316 x^{2} + 1184 x + 1608$	$2$	$2$	$3$	$9$	$A_4\times C_2$ (as 6T6)	$3$	$1$	$[3]$	$[2, 2, 3]^{3}$	$t^{3} + t + 1$	$x^{2} + \left(4 t^{2} + 4 t\right) x + 12 t^{2} + 4 t + 2$	$[2, 0]$	$[1]$
2.6.10.1	$6$	$x^{6} + 2 x^{5} + 4 x^{2} + 4 x + 2$	$2$	$6$	$1$	$10$	$S_4$ (as 6T7)	$2$	$3$	$[8/3]$	$[8/3, 8/3]_{3}^{2}$	$t + 1$	$x^{6} + 2 x^{5} + 4 x^{2} + 4 x + 2$	$[5, 0]$	$[1, 2]$
2.6.10.2	$6$	$x^{6} + 2 x^{5} + 4 x^{4} + 4 x + 2$	$2$	$6$	$1$	$10$	$S_4$ (as 6T7)	$2$	$3$	$[8/3]$	$[8/3, 8/3]_{3}^{2}$	$t + 1$	$x^{6} + 2 x^{5} + 4 x^{4} + 4 x + 2$	$[5, 0]$	$[1, 2]$
2.6.10.3	$6$	$x^{6} + 2 x^{5} + 4 x + 2$	$2$	$6$	$1$	$10$	$S_4$ (as 6T8)	$2$	$3$	$[8/3]$	$[8/3, 8/3]_{3}^{2}$	$t + 1$	$x^{6} + 2 x^{5} + 4 x + 2$	$[5, 0]$	$[1, 2]$
2.6.10.4	$6$	$x^{6} + 2 x^{5} + 4 x^{3} + 4 x + 2$	$2$	$6$	$1$	$10$	$S_4$ (as 6T8)	$2$	$3$	$[8/3]$	$[8/3, 8/3]_{3}^{2}$	$t + 1$	$x^{6} + 2 x^{5} + 4 x^{3} + 4 x + 2$	$[5, 0]$	$[1, 2]$
2.6.10.5	$6$	$x^{6} + 2 x^{5} + 4 x^{3} + 4 x^{2} + 2$	$2$	$6$	$1$	$10$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[8/3]$	$[2, 8/3, 8/3]_{3}^{2}$	$t + 1$	$x^{6} + 2 x^{5} + 4 x^{3} + 4 x^{2} + 2$	$[5, 0]$	$[1, 2]$
2.6.10.6	$6$	$x^{6} + 2 x^{5} + 4 x^{2} + 2$	$2$	$6$	$1$	$10$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[8/3]$	$[2, 8/3, 8/3]_{3}^{2}$	$t + 1$	$x^{6} + 2 x^{5} + 4 x^{2} + 2$	$[5, 0]$	$[1, 2]$
2.6.10.7	$6$	$x^{6} + 2 x^{5} + 4 x^{3} + 2$	$2$	$6$	$1$	$10$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[8/3]$	$[2, 8/3, 8/3]_{3}^{2}$	$t + 1$	$x^{6} + 2 x^{5} + 4 x^{3} + 2$	$[5, 0]$	$[1, 2]$
2.6.10.8	$6$	$x^{6} + 2 x^{5} + 2$	$2$	$6$	$1$	$10$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[8/3]$	$[2, 8/3, 8/3]_{3}^{2}$	$t + 1$	$x^{6} + 2 x^{5} + 2$	$[5, 0]$	$[1, 2]$
2.6.11.1	$6$	$x^{6} + 4 x^{3} + 2$	$2$	$6$	$1$	$11$	$D_{6}$ (as 6T3)	$2$	$3$	$[3]$	$[3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{3} + 2$	$[6, 0]$	$[1, 2]$
2.6.11.2	$6$	$x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 2$	$2$	$6$	$1$	$11$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[3]$	$[4/3, 4/3, 3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 2$	$[6, 0]$	$[1, 2]$
2.6.11.3	$6$	$x^{6} + 4 x^{2} + 4 x + 10$	$2$	$6$	$1$	$11$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[3]$	$[8/3, 8/3, 3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{2} + 4 x + 10$	$[6, 0]$	$[1, 2]$
2.6.11.4	$6$	$x^{6} + 4 x^{4} + 4 x^{2} + 4 x + 2$	$2$	$6$	$1$	$11$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[3]$	$[8/3, 8/3, 3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{4} + 4 x^{2} + 4 x + 2$	$[6, 0]$	$[1, 2]$
2.6.11.5	$6$	$x^{6} + 4 x^{3} + 10$	$2$	$6$	$1$	$11$	$D_{6}$ (as 6T3)	$2$	$3$	$[3]$	$[3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{3} + 10$	$[6, 0]$	$[1, 2]$
2.6.11.6	$6$	$x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 10$	$2$	$6$	$1$	$11$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[3]$	$[4/3, 4/3, 3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{5} + 4 x^{4} + 4 x^{3} + 4 x^{2} + 10$	$[6, 0]$	$[1, 2]$
2.6.11.7	$6$	$x^{6} + 4 x^{2} + 4 x + 2$	$2$	$6$	$1$	$11$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[3]$	$[8/3, 8/3, 3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{2} + 4 x + 2$	$[6, 0]$	$[1, 2]$
2.6.11.8	$6$	$x^{6} + 4 x^{5} + 4 x^{2} + 4 x + 2$	$2$	$6$	$1$	$11$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[3]$	$[8/3, 8/3, 3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{5} + 4 x^{2} + 4 x + 2$	$[6, 0]$	$[1, 2]$
2.6.11.9	$6$	$x^{6} + 4 x^{5} + 4 x^{2} + 2$	$2$	$6$	$1$	$11$	$D_{6}$ (as 6T3)	$2$	$3$	$[3]$	$[3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{5} + 4 x^{2} + 2$	$[6, 0]$	$[1, 2]$
2.6.11.10	$6$	$x^{6} + 4 x^{2} + 2$	$2$	$6$	$1$	$11$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[3]$	$[4/3, 4/3, 3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{2} + 2$	$[6, 0]$	$[1, 2]$
2.6.11.11	$6$	$x^{6} + 4 x^{4} + 4 x + 14$	$2$	$6$	$1$	$11$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[3]$	$[8/3, 8/3, 3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{4} + 4 x + 14$	$[6, 0]$	$[1, 2]$
2.6.11.12	$6$	$x^{6} + 4 x^{4} + 4 x^{3} + 4 x + 10$	$2$	$6$	$1$	$11$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[3]$	$[8/3, 8/3, 3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{4} + 4 x^{3} + 4 x + 10$	$[6, 0]$	$[1, 2]$
2.6.11.13	$6$	$x^{6} + 10$	$2$	$6$	$1$	$11$	$D_{6}$ (as 6T3)	$2$	$3$	$[3]$	$[3]_{3}^{2}$	$t + 1$	$x^{6} + 10$	$[6, 0]$	$[1, 2]$
2.6.11.14	$6$	$x^{6} + 4 x^{2} + 10$	$2$	$6$	$1$	$11$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[3]$	$[4/3, 4/3, 3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{2} + 10$	$[6, 0]$	$[1, 2]$
2.6.11.15	$6$	$x^{6} + 4 x^{4} + 4 x + 6$	$2$	$6$	$1$	$11$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[3]$	$[8/3, 8/3, 3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{4} + 4 x + 6$	$[6, 0]$	$[1, 2]$
2.6.11.16	$6$	$x^{6} + 4 x^{4} + 4 x^{3} + 4 x + 2$	$2$	$6$	$1$	$11$	$S_4\times C_2$ (as 6T11)	$2$	$3$	$[3]$	$[8/3, 8/3, 3]_{3}^{2}$	$t + 1$	$x^{6} + 4 x^{4} + 4 x^{3} + 4 x + 2$	$[6, 0]$	$[1, 2]$

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.