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 (28 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
13.12.0.1	$12$	x¹² + x⁸ + 5x⁷ + 8x⁶ + 11x⁵ + 3x⁴ + x³ + x² + 4x + 2	$13$	$1$	$12$	$0$	$C_{12}$ (as 12T1)	$12$	$1$	$[\ ]$	$[\ ]^{12}$	$[0]$	$[]$
13.12.6.1	$12$	x¹² + 780x¹¹ + 253578x¹⁰ + 43990720x⁹ + 4297346257x⁸ + 224493831662x⁷ + 4938918346310x⁶ + 2961720498866x⁵ + 3005850529646x⁴ + 51307643736852x³ + 70292613843513x² + 65587287977710x + 15475747398037	$13$	$2$	$6$	$6$	$C_6\times C_2$ (as 12T2)	$6$	$2$	$[\ ]$	$[\ ]_{2}^{6}$	$[0]$	$[1]$
13.12.6.2	$12$	x¹² - 21970x⁶ + 314171x⁴ - 4084223x² + 9653618	$13$	$2$	$6$	$6$	$C_{12}$ (as 12T1)	$6$	$2$	$[\ ]$	$[\ ]_{2}^{6}$	$[0]$	$[1]$
13.12.8.1	$12$	x¹² + 9x¹⁰ + 88x⁹ + 33x⁸ + 216x⁷ - 1299x⁶ - 78x⁵ - 1797x⁴ - 15494x³ + 21687x² - 41586x + 201846	$13$	$3$	$4$	$8$	$C_{12}$ (as 12T1)	$4$	$3$	$[\ ]$	$[\ ]_{3}^{4}$	$[0]$	$[1]$
13.12.8.2	$12$	x¹² + 507x⁶ - 26364x³ + 57122	$13$	$3$	$4$	$8$	$C_{12}$ (as 12T1)	$4$	$3$	$[\ ]$	$[\ ]_{3}^{4}$	$[0]$	$[1]$
13.12.8.3	$12$	x¹² - 78x⁹ + 2197x⁶ + 290004x³ + 114244	$13$	$3$	$4$	$8$	$C_{12}$ (as 12T1)	$4$	$3$	$[\ ]$	$[\ ]_{3}^{4}$	$[0]$	$[1]$
13.12.9.1	$12$	x¹² - 52x⁸ + 676x⁴ + 265837	$13$	$4$	$3$	$9$	$C_{12}$ (as 12T1)	$3$	$4$	$[\ ]$	$[\ ]_{4}^{3}$	$[0]$	$[1]$
13.12.9.2	$12$	x¹² + 8x¹⁰ + 44x⁹ + 63x⁸ + 264x⁷ + 550x⁶ - 6336x⁵ + 3843x⁴ + 4532x³ + 46454x² + 30668x + 30982	$13$	$4$	$3$	$9$	$C_{12}$ (as 12T1)	$3$	$4$	$[\ ]$	$[\ ]_{4}^{3}$	$[0]$	$[1]$
13.12.9.3	$12$	x¹² + 338x⁴ - 24167	$13$	$4$	$3$	$9$	$C_{12}$ (as 12T1)	$3$	$4$	$[\ ]$	$[\ ]_{4}^{3}$	$[0]$	$[1]$
13.12.9.4	$12$	x¹² + 78x⁸ + 42926x⁴ - 31085353	$13$	$4$	$3$	$9$	$C_{12}$ (as 12T1)	$3$	$4$	$[\ ]$	$[\ ]_{4}^{3}$	$[0]$	$[1]$
13.12.10.1	$12$	x¹² + 72x¹¹ + 2172x¹⁰ + 35280x⁹ + 328380x⁸ + 1703232x⁷ + 4282170x⁶ + 3407400x⁵ + 1340820x⁴ + 712800x³ + 3855192x² + 18082080x + 35650393	$13$	$6$	$2$	$10$	$C_6\times C_2$ (as 12T2)	$2$	$6$	$[\ ]$	$[\ ]_{6}^{2}$	$[0]$	$[1]$
13.12.10.2	$12$	x¹² + 130x⁶ - 1521	$13$	$6$	$2$	$10$	$C_6\times C_2$ (as 12T2)	$2$	$6$	$[\ ]$	$[\ ]_{6}^{2}$	$[0]$	$[1]$
13.12.10.3	$12$	x¹² - 1508x⁶ - 6084	$13$	$6$	$2$	$10$	$C_6\times C_2$ (as 12T2)	$2$	$6$	$[\ ]$	$[\ ]_{6}^{2}$	$[0]$	$[1]$
13.12.10.4	$12$	x¹² - 156x⁶ + 338	$13$	$6$	$2$	$10$	$C_{12}$ (as 12T1)	$2$	$6$	$[\ ]$	$[\ ]_{6}^{2}$	$[0]$	$[1]$
13.12.10.5	$12$	x¹² - 1586x⁶ - 198575	$13$	$6$	$2$	$10$	$C_{12}$ (as 12T1)	$2$	$6$	$[\ ]$	$[\ ]_{6}^{2}$	$[0]$	$[1]$
13.12.10.6	$12$	x¹² - 1716x⁶ - 91260	$13$	$6$	$2$	$10$	$C_{12}$ (as 12T1)	$2$	$6$	$[\ ]$	$[\ ]_{6}^{2}$	$[0]$	$[1]$
13.12.11.1	$12$	x¹² + 156	$13$	$12$	$1$	$11$	$C_{12}$ (as 12T1)	$1$	$12$	$[\ ]$	$[\ ]_{12}$	$[0]$	$[1]$
13.12.11.10	$12$	x¹² + 143	$13$	$12$	$1$	$11$	$C_{12}$ (as 12T1)	$1$	$12$	$[\ ]$	$[\ ]_{12}$	$[0]$	$[1]$
13.12.11.11	$12$	x¹² + 65	$13$	$12$	$1$	$11$	$C_{12}$ (as 12T1)	$1$	$12$	$[\ ]$	$[\ ]_{12}$	$[0]$	$[1]$
13.12.11.12	$12$	x¹² + 91	$13$	$12$	$1$	$11$	$C_{12}$ (as 12T1)	$1$	$12$	$[\ ]$	$[\ ]_{12}$	$[0]$	$[1]$
13.12.11.2	$12$	x¹² + 117	$13$	$12$	$1$	$11$	$C_{12}$ (as 12T1)	$1$	$12$	$[\ ]$	$[\ ]_{12}$	$[0]$	$[1]$
13.12.11.3	$12$	x¹² + 130	$13$	$12$	$1$	$11$	$C_{12}$ (as 12T1)	$1$	$12$	$[\ ]$	$[\ ]_{12}$	$[0]$	$[1]$
13.12.11.4	$12$	x¹² + 13	$13$	$12$	$1$	$11$	$C_{12}$ (as 12T1)	$1$	$12$	$[\ ]$	$[\ ]_{12}$	$[0]$	$[1]$
13.12.11.5	$12$	x¹² + 52	$13$	$12$	$1$	$11$	$C_{12}$ (as 12T1)	$1$	$12$	$[\ ]$	$[\ ]_{12}$	$[0]$	$[1]$
13.12.11.6	$12$	x¹² + 39	$13$	$12$	$1$	$11$	$C_{12}$ (as 12T1)	$1$	$12$	$[\ ]$	$[\ ]_{12}$	$[0]$	$[1]$
13.12.11.7	$12$	x¹² + 26	$13$	$12$	$1$	$11$	$C_{12}$ (as 12T1)	$1$	$12$	$[\ ]$	$[\ ]_{12}$	$[0]$	$[1]$
13.12.11.8	$12$	x¹² + 104	$13$	$12$	$1$	$11$	$C_{12}$ (as 12T1)	$1$	$12$	$[\ ]$	$[\ ]_{12}$	$[0]$	$[1]$
13.12.11.9	$12$	x¹² + 78	$13$	$12$	$1$	$11$	$C_{12}$ (as 12T1)	$1$	$12$	$[\ ]$	$[\ ]_{12}$	$[0]$	$[1]$

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