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	Num. automorphisms

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

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Results (22 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
2.4.2.8a1.1	$8$	$( x^{4} + x + 1 )^{2} + 2 ( x^{4} + x + 1 ) + 2$	$2$	$4$	$2$	$8$	$C_4\times C_2$ (as 8T2)	$4$	$1$	$[2]$	$[1]$	$[2]^{4}$	$[1]^{4}$	$[\ ]$	$[\ ]$	$8$	$t^{4} + t + 1$	$x^{2} + 2 x + 2$	$[1, 0]$	$[1]$	$z + 1$	$[1, 2]$
2.4.2.8a1.2	$8$	$( x^{4} + x + 1 )^{2} + 2 ( x^{4} + x + 1 ) + 4 x^{3} + 2$	$2$	$4$	$2$	$8$	$C_8$ (as 8T1)	$4$	$1$	$[2]$	$[1]$	$[2]^{4}$	$[1]^{4}$	$[\ ]$	$[\ ]$	$8$	$t^{4} + t + 1$	$x^{2} + 2 x + 4 t^{3} + 2$	$[1, 0]$	$[1]$	$z + 1$	$[1, 4]$
2.4.2.8a2.1	$8$	$( x^{4} + x + 1 )^{2} + 2 x ( x^{4} + x + 1 ) + 2$	$2$	$4$	$2$	$8$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$4$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2]^{4}$	$[1,1,1,1]^{4}$	$[2,2,2]$	$[1,1,1]$	$2$	$t^{4} + t + 1$	$x^{2} + \left(2 t^{2} + 2\right) x + 6$	$[1, 0]$	$[1]$	$z + t$	$[1, 3]$
2.4.2.8a2.2	$8$	$( x^{4} + x + 1 )^{2} + 2 x ( x^{4} + x + 1 ) + 6$	$2$	$4$	$2$	$8$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$4$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2]^{4}$	$[1,1,1,1]^{4}$	$[2,2,2]$	$[1,1,1]$	$2$	$t^{4} + t + 1$	$x^{2} + \left(2 t^{2} + 2\right) x + 2$	$[1, 0]$	$[1]$	$z + t$	$[1, 3]$
2.4.2.8a3.1	$8$	$( x^{4} + x + 1 )^{2} + \left(2 x^{2} + 2 x\right) ( x^{4} + x + 1 ) + 2$	$2$	$4$	$2$	$8$	$C_2^2:C_4$ (as 8T10)	$4$	$1$	$[2]$	$[1]$	$[2, 2]^{4}$	$[1,1]^{4}$	$[2]$	$[1]$	$4$	$t^{4} + t + 1$	$x^{2} + \left(2 t^{2} + 2 t + 2\right) x + 2$	$[1, 0]$	$[1]$	$z + (t^2 + t)$	$[1, 3]$
2.4.2.8a3.2	$8$	$( x^{4} + x + 1 )^{2} + \left(2 x^{2} + 2 x\right) ( x^{4} + x + 1 ) + 4 x + 2$	$2$	$4$	$2$	$8$	$C_8:C_2$ (as 8T7)	$4$	$1$	$[2]$	$[1]$	$[2, 2]^{4}$	$[1,1]^{4}$	$[2]$	$[1]$	$4$	$t^{4} + t + 1$	$x^{2} + \left(2 t^{2} + 2 t + 2\right) x + 4 t + 2$	$[1, 0]$	$[1]$	$z + (t^2 + t)$	$[1, 3]$
2.4.2.8a4.1	$8$	$( x^{4} + x + 1 )^{2} + 2 x^{3} ( x^{4} + x + 1 ) + 2$	$2$	$4$	$2$	$8$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$4$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2]^{4}$	$[1,1,1,1]^{4}$	$[2,2,2]$	$[1,1,1]$	$2$	$t^{4} + t + 1$	$x^{2} + \left(2 t^{3} + 2 t\right) x + 6$	$[1, 0]$	$[1]$	$z + t^3$	$[1, 3]$
2.4.2.8a4.2	$8$	$( x^{4} + x + 1 )^{2} + 2 x^{3} ( x^{4} + x + 1 ) + 6$	$2$	$4$	$2$	$8$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$4$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2]^{4}$	$[1,1,1,1]^{4}$	$[2,2,2]$	$[1,1,1]$	$2$	$t^{4} + t + 1$	$x^{2} + \left(2 t^{3} + 2 t\right) x + 2$	$[1, 0]$	$[1]$	$z + t^3$	$[1, 3]$
2.4.2.8a5.1	$8$	$( x^{4} + x + 1 )^{2} + \left(2 x^{3} + 2 x + 2\right) ( x^{4} + x + 1 ) + 2$	$2$	$4$	$2$	$8$	$(C_8:C_2):C_2$ (as 8T16)	$4$	$1$	$[2]$	$[1]$	$[2, 2, 2]^{4}$	$[1,1,1]^{4}$	$[2,2]$	$[1,1]$	$2$	$t^{4} + t + 1$	$x^{2} + \left(2 t^{3} + 2 t + 2\right) x + 4 t^{2} + 2$	$[1, 0]$	$[1]$	$z + (t^3 + t + 1)$	$[1, 3]$
2.4.2.8a5.2	$8$	$( x^{4} + x + 1 )^{2} + \left(2 x^{3} + 2 x + 2\right) ( x^{4} + x + 1 ) + 4 x^{2} + 2$	$2$	$4$	$2$	$8$	$C_2^3: C_4$ (as 8T20)	$4$	$1$	$[2]$	$[1]$	$[2, 2, 2]^{4}$	$[1,1,1]^{4}$	$[2,2]$	$[1,1]$	$2$	$t^{4} + t + 1$	$x^{2} + \left(2 t^{3} + 2 t^{2} + 2\right) x + 2$	$[1, 0]$	$[1]$	$z + (t^3 + t + 1)$	$[1, 3]$
2.4.2.12a1.1	$8$	$( x^{4} + x + 1 )^{2} + 2$	$2$	$4$	$2$	$12$	$C_4\times C_2$ (as 8T2)	$4$	$1$	$[3]$	$[2]$	$[3]^{4}$	$[2]^{4}$	$[\ ]$	$[\ ]$	$8$	$t^{4} + t + 1$	$x^{2} + 2$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.4.2.12a1.2	$8$	$( x^{4} + x + 1 )^{2} + 8 x^{3} + 2$	$2$	$4$	$2$	$12$	$C_8$ (as 8T1)	$4$	$1$	$[3]$	$[2]$	$[3]^{4}$	$[2]^{4}$	$[\ ]$	$[\ ]$	$8$	$t^{4} + t + 1$	$x^{2} + 8 t^{3} + 2$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.4.2.12a1.3	$8$	$( x^{4} + x + 1 )^{2} + 4 x^{3} ( x^{4} + x + 1 ) + 2$	$2$	$4$	$2$	$12$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$4$	$1$	$[3]$	$[2]$	$[2, 2, 2, 3]^{4}$	$[1,1,1,2]^{4}$	$[2,2,2]$	$[1,1,1]$	$2$	$t^{4} + t + 1$	$x^{2} + \left(4 t^{3} + 4 t^{2} + 4 t + 4\right) x + 2$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.4.2.12a1.4	$8$	$( x^{4} + x + 1 )^{2} + 4 x^{3} ( x^{4} + x + 1 ) + 8 x^{3} + 2$	$2$	$4$	$2$	$12$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$4$	$1$	$[3]$	$[2]$	$[2, 2, 2, 3]^{4}$	$[1,1,1,2]^{4}$	$[2,2,2]$	$[1,1,1]$	$2$	$t^{4} + t + 1$	$x^{2} + \left(4 t^{3} + 4 t^{2} + 4 t + 4\right) x + 8 t^{3} + 2$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.4.2.12a1.5	$8$	$( x^{4} + x + 1 )^{2} + 4 x ( x^{4} + x + 1 ) + 2$	$2$	$4$	$2$	$12$	$C_2^3: C_4$ (as 8T20)	$4$	$1$	$[3]$	$[2]$	$[2, 2, 3]^{4}$	$[1,1,2]^{4}$	$[2,2]$	$[1,1]$	$2$	$t^{4} + t + 1$	$x^{2} + \left(4 t^{2} + 4\right) x + 2$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.4.2.12a1.6	$8$	$( x^{4} + x + 1 )^{2} + 4 x ( x^{4} + x + 1 ) + 8 x^{3} + 2$	$2$	$4$	$2$	$12$	$(C_8:C_2):C_2$ (as 8T16)	$4$	$1$	$[3]$	$[2]$	$[2, 2, 3]^{4}$	$[1,1,2]^{4}$	$[2,2]$	$[1,1]$	$2$	$t^{4} + t + 1$	$x^{2} + \left(4 t^{2} + 4\right) x + 8 t^{3} + 2$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.4.2.12a1.7	$8$	$( x^{4} + x + 1 )^{2} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 ) + 2$	$2$	$4$	$2$	$12$	$C_2^2:C_4$ (as 8T10)	$4$	$1$	$[3]$	$[2]$	$[2, 3]^{4}$	$[1,2]^{4}$	$[2]$	$[1]$	$4$	$t^{4} + t + 1$	$x^{2} + \left(4 t^{2} + 4 t + 4\right) x + 2$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.4.2.12a1.8	$8$	$( x^{4} + x + 1 )^{2} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$	$2$	$4$	$2$	$12$	$C_8:C_2$ (as 8T7)	$4$	$1$	$[3]$	$[2]$	$[2, 3]^{4}$	$[1,2]^{4}$	$[2]$	$[1]$	$4$	$t^{4} + t + 1$	$x^{2} + \left(4 t^{2} + 4 t + 4\right) x + 8 t^{3} + 2$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.4.2.12a1.9	$8$	$( x^{4} + x + 1 )^{2} + 4 ( x^{4} + x + 1 ) + 2$	$2$	$4$	$2$	$12$	$C_4\times C_2$ (as 8T2)	$4$	$1$	$[3]$	$[2]$	$[3]^{4}$	$[2]^{4}$	$[\ ]$	$[\ ]$	$8$	$t^{4} + t + 1$	$x^{2} + 4 x + 2$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.4.2.12a1.10	$8$	$( x^{4} + x + 1 )^{2} + 4 ( x^{4} + x + 1 ) + 8 x^{3} + 2$	$2$	$4$	$2$	$12$	$C_8$ (as 8T1)	$4$	$1$	$[3]$	$[2]$	$[3]^{4}$	$[2]^{4}$	$[\ ]$	$[\ ]$	$8$	$t^{4} + t + 1$	$x^{2} + 4 x + 8 t^{3} + 2$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.4.2.12a1.11	$8$	$( x^{4} + x + 1 )^{2} + \left(4 x^{3} + 4\right) ( x^{4} + x + 1 ) + 2$	$2$	$4$	$2$	$12$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$4$	$1$	$[3]$	$[2]$	$[2, 2, 2, 3]^{4}$	$[1,1,1,2]^{4}$	$[2,2,2]$	$[1,1,1]$	$2$	$t^{4} + t + 1$	$x^{2} + \left(4 t^{3} + 4 t^{2} + 4 t\right) x + 2$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.4.2.12a1.12	$8$	$( x^{4} + x + 1 )^{2} + \left(4 x^{3} + 4\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$	$2$	$4$	$2$	$12$	$((C_8 : C_2):C_2):C_2$ (as 8T27)	$4$	$1$	$[3]$	$[2]$	$[2, 2, 2, 3]^{4}$	$[1,1,1,2]^{4}$	$[2,2,2]$	$[1,1,1]$	$2$	$t^{4} + t + 1$	$x^{2} + \left(4 t^{3} + 4 t^{2} + 4 t\right) x + 8 t^{3} + 2$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$

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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.