LMFDB - p-adic field search results

The results below are complete, since the LMFDB contains all p-adic fields of degree at most 23 and residue characteristic at most 199

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Degree $n$	Residue characteristic $p$	Discriminant exponent $c$	Galois group

Ramification index $e$	Residue field degree $f$	Galois tame degree $t$	Galois unram. degree $u$

Num. automorphisms	Inertia subgroup	Indices	Assoc. Inertia	Jump Set

Top Artin slope	Galois Artin	Visible Artin	Wild inertia subgroup

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Results (20 matches)

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Label	$n$	Polynomial	$p$	$f$	$e$	$c$	Galois group	$u$	$t$	Visible Artin slopes	Visible Swan slopes	Artin slope content	Swan slope content	Hidden Artin slopes	Hidden Swan slopes	$\#\Aut(K/\Q_p)$	Unram. Ext.	Eisen. Poly.	Ind. of Insep.	Assoc. Inertia	Resid. Poly	Jump Set
2.1.4.11a1.1	$4$	$x^{4} + 8 x^{3} + 2$	$2$	$1$	$4$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 4]$	$[2,3]$	$[2, 3, 4]$	$[1,2,3]$	$[2]$	$[1]$	$2$	$t + 1$	$x^{4} + 8 x^{3} + 2$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.2	$4$	$x^{4} + 8 x^{3} + 18$	$2$	$1$	$4$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 4]$	$[2,3]$	$[2, 3, 4]$	$[1,2,3]$	$[2]$	$[1]$	$2$	$t + 1$	$x^{4} + 8 x^{3} + 18$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.3	$4$	$x^{4} + 8 x + 18$	$2$	$1$	$4$	$11$	$D_{4}$ (as 4T3)	$2$	$1$	$[3, 4]$	$[2,3]$	$[3, 4]^{2}$	$[2,3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$2$	$t + 1$	$x^{4} + 8 x + 18$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.4	$4$	$x^{4} + 8 x^{3} + 8 x + 18$	$2$	$1$	$4$	$11$	$D_{4}$ (as 4T3)	$2$	$1$	$[3, 4]$	$[2,3]$	$[3, 4]^{2}$	$[2,3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$2$	$t + 1$	$x^{4} + 8 x^{3} + 8 x + 18$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.5	$4$	$x^{4} + 8 x^{3} + 10$	$2$	$1$	$4$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 4]$	$[2,3]$	$[2, 3, 4]$	$[1,2,3]$	$[2]$	$[1]$	$2$	$t + 1$	$x^{4} + 8 x^{3} + 10$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.6	$4$	$x^{4} + 8 x^{3} + 26$	$2$	$1$	$4$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 4]$	$[2,3]$	$[2, 3, 4]$	$[1,2,3]$	$[2]$	$[1]$	$2$	$t + 1$	$x^{4} + 8 x^{3} + 26$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.7	$4$	$x^{4} + 8 x + 26$	$2$	$1$	$4$	$11$	$D_{4}$ (as 4T3)	$2$	$1$	$[3, 4]$	$[2,3]$	$[3, 4]^{2}$	$[2,3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$2$	$t + 1$	$x^{4} + 8 x + 26$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.8	$4$	$x^{4} + 8 x^{3} + 8 x + 26$	$2$	$1$	$4$	$11$	$D_{4}$ (as 4T3)	$2$	$1$	$[3, 4]$	$[2,3]$	$[3, 4]^{2}$	$[2,3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$2$	$t + 1$	$x^{4} + 8 x^{3} + 8 x + 26$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.9	$4$	$x^{4} + 4 x^{2} + 2$	$2$	$1$	$4$	$11$	$C_4$ (as 4T1)	$1$	$1$	$[3, 4]$	$[2,3]$	$[3, 4]$	$[2,3]$	$[\ ]$	$[\ ]$	$4$	$t + 1$	$x^{4} + 4 x^{2} + 2$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.10	$4$	$x^{4} + 4 x^{2} + 18$	$2$	$1$	$4$	$11$	$C_4$ (as 4T1)	$1$	$1$	$[3, 4]$	$[2,3]$	$[3, 4]$	$[2,3]$	$[\ ]$	$[\ ]$	$4$	$t + 1$	$x^{4} + 4 x^{2} + 18$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.11	$4$	$x^{4} + 8 x^{3} + 4 x^{2} + 2$	$2$	$1$	$4$	$11$	$C_4$ (as 4T1)	$1$	$1$	$[3, 4]$	$[2,3]$	$[3, 4]$	$[2,3]$	$[\ ]$	$[\ ]$	$4$	$t + 1$	$x^{4} + 8 x^{3} + 4 x^{2} + 2$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.12	$4$	$x^{4} + 8 x^{3} + 4 x^{2} + 18$	$2$	$1$	$4$	$11$	$C_4$ (as 4T1)	$1$	$1$	$[3, 4]$	$[2,3]$	$[3, 4]$	$[2,3]$	$[\ ]$	$[\ ]$	$4$	$t + 1$	$x^{4} + 8 x^{3} + 4 x^{2} + 18$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.13	$4$	$x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 2$	$2$	$1$	$4$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 4]$	$[2,3]$	$[2, 3, 4]$	$[1,2,3]$	$[2]$	$[1]$	$2$	$t + 1$	$x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 2$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.14	$4$	$x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 18$	$2$	$1$	$4$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 4]$	$[2,3]$	$[2, 3, 4]$	$[1,2,3]$	$[2]$	$[1]$	$2$	$t + 1$	$x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 18$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.15	$4$	$x^{4} + 4 x^{2} + 10$	$2$	$1$	$4$	$11$	$C_4$ (as 4T1)	$1$	$1$	$[3, 4]$	$[2,3]$	$[3, 4]$	$[2,3]$	$[\ ]$	$[\ ]$	$4$	$t + 1$	$x^{4} + 4 x^{2} + 10$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.16	$4$	$x^{4} + 4 x^{2} + 26$	$2$	$1$	$4$	$11$	$C_4$ (as 4T1)	$1$	$1$	$[3, 4]$	$[2,3]$	$[3, 4]$	$[2,3]$	$[\ ]$	$[\ ]$	$4$	$t + 1$	$x^{4} + 4 x^{2} + 26$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.17	$4$	$x^{4} + 8 x^{3} + 4 x^{2} + 10$	$2$	$1$	$4$	$11$	$C_4$ (as 4T1)	$1$	$1$	$[3, 4]$	$[2,3]$	$[3, 4]$	$[2,3]$	$[\ ]$	$[\ ]$	$4$	$t + 1$	$x^{4} + 8 x^{3} + 4 x^{2} + 10$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.18	$4$	$x^{4} + 8 x^{3} + 4 x^{2} + 26$	$2$	$1$	$4$	$11$	$C_4$ (as 4T1)	$1$	$1$	$[3, 4]$	$[2,3]$	$[3, 4]$	$[2,3]$	$[\ ]$	$[\ ]$	$4$	$t + 1$	$x^{4} + 8 x^{3} + 4 x^{2} + 26$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.19	$4$	$x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 10$	$2$	$1$	$4$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 4]$	$[2,3]$	$[2, 3, 4]$	$[1,2,3]$	$[2]$	$[1]$	$2$	$t + 1$	$x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 10$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.20	$4$	$x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 26$	$2$	$1$	$4$	$11$	$D_{4}$ (as 4T3)	$1$	$1$	$[3, 4]$	$[2,3]$	$[2, 3, 4]$	$[1,2,3]$	$[2]$	$[1]$	$2$	$t + 1$	$x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 26$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$

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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.