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 (30 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
2.5.2.10a1.1	$10$	$( x^{5} + x^{2} + 1 )^{2} + 2 ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$10$	$C_{10}$ (as 10T1)	$5$	$1$	$[2]$	$[1]$	$[2]^{5}$	$[1]^{5}$	$[\ ]$	$[\ ]$	$10$	$t^{5} + t^{2} + 1$	$x^{2} + 2 x + 2$	$[1, 0]$	$[1]$	$z + t^4$	$[1, 2]$
2.5.2.10a1.2	$10$	$( x^{5} + x^{2} + 1 )^{2} + 2 ( x^{5} + x^{2} + 1 ) + 6$	$2$	$5$	$2$	$10$	$C_{10}$ (as 10T1)	$5$	$1$	$[2]$	$[1]$	$[2]^{5}$	$[1]^{5}$	$[\ ]$	$[\ ]$	$10$	$t^{5} + t^{2} + 1$	$x^{2} + 2 x + 6$	$[1, 0]$	$[1]$	$z + t^4$	$[1, 4]$
2.5.2.10a2.1	$10$	$( x^{5} + x^{2} + 1 )^{2} + 2 x ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$10$	$C_2^4 : C_5$ (as 10T8)	$5$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2]^{5}$	$[1,1,1,1]^{5}$	$[2,2,2]$	$[1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(2 t^{4} + 2 t^{3} + 2 t + 2\right) x + 2$	$[1, 0]$	$[1]$	$z + (t^2 + 1)$	$[1, 3]$
2.5.2.10a2.2	$10$	$( x^{5} + x^{2} + 1 )^{2} + 2 x ( x^{5} + x^{2} + 1 ) + 4 x^{2} + 2$	$2$	$5$	$2$	$10$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$10$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2]^{10}$	$[1,1,1,1]^{10}$	$[2,2,2]^{2}$	$[1,1,1]^{2}$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + 2 t x + 4 t^{2} + 2$	$[1, 0]$	$[1]$	$z + (t^2 + 1)$	$[1, 3]$
2.5.2.10a3.1	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{2} + 2\right) ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$10$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2, 2]^{5}$	$[1,1,1,1,1]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(2 t^{4} + 2\right) x + 2$	$[1, 0]$	$[1]$	$z + (t^4 + t^3 + t)$	$[1, 3]$
2.5.2.10a3.2	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{2} + 2\right) ( x^{5} + x^{2} + 1 ) + 6$	$2$	$5$	$2$	$10$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2, 2]^{5}$	$[1,1,1,1,1]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(2 t^{4} + 2\right) x + 6$	$[1, 0]$	$[1]$	$z + (t^4 + t^3 + t)$	$[1, 3]$
2.5.2.10a4.1	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{2} + 2 x + 2\right) ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$10$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2, 2]^{5}$	$[1,1,1,1,1]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(2 t^{4} + 2 t^{2} + 2\right) x + 6$	$[1, 0]$	$[1]$	$z + (t^4 + t^3 + t^2 + t + 1)$	$[1, 3]$
2.5.2.10a4.2	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{2} + 2 x + 2\right) ( x^{5} + x^{2} + 1 ) + 6$	$2$	$5$	$2$	$10$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2, 2]^{5}$	$[1,1,1,1,1]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(2 t^{4} + 2 t^{2} + 2\right) x + 2$	$[1, 0]$	$[1]$	$z + (t^4 + t^3 + t^2 + t + 1)$	$[1, 3]$
2.5.2.10a5.1	$10$	$( x^{5} + x^{2} + 1 )^{2} + 2 x^{3} ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$10$	$C_2^4 : C_5$ (as 10T8)	$5$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2]^{5}$	$[1,1,1,1]^{5}$	$[2,2,2]$	$[1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(2 t^{4} + 2 t + 2\right) x + 2$	$[1, 0]$	$[1]$	$z + (t^4 + t^2)$	$[1, 3]$
2.5.2.10a5.2	$10$	$( x^{5} + x^{2} + 1 )^{2} + 2 x^{3} ( x^{5} + x^{2} + 1 ) + 4 x + 2$	$2$	$5$	$2$	$10$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$10$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2]^{10}$	$[1,1,1,1]^{10}$	$[2,2,2]^{2}$	$[1,1,1]^{2}$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(2 t^{3} + 2 t^{2} + 2 t\right) x + 4 t^{2} + 2$	$[1, 0]$	$[1]$	$z + (t^4 + t^2)$	$[1, 3]$
2.5.2.10a6.1	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{4} + 2 x^{2}\right) ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$10$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2, 2]^{5}$	$[1,1,1,1,1]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(2 t^{4} + 2 t^{3} + 2 t^{2} + 2\right) x + 2$	$[1, 0]$	$[1]$	$z + (t^2 + t + 1)$	$[1, 3]$
2.5.2.10a6.2	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{4} + 2 x^{2}\right) ( x^{5} + x^{2} + 1 ) + 6$	$2$	$5$	$2$	$10$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2, 2]^{5}$	$[1,1,1,1,1]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(2 t^{4} + 2 t^{3} + 2 t^{2} + 2\right) x + 6$	$[1, 0]$	$[1]$	$z + (t^2 + t + 1)$	$[1, 3]$
2.5.2.10a7.1	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{4} + 2 x^{3} + 2 x^{2} + 2 x + 2\right) ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$10$	$C_2^4 : C_5$ (as 10T8)	$5$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2]^{5}$	$[1,1,1,1]^{5}$	$[2,2,2]$	$[1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(2 t^{4} + 2 t^{3} + 2 t^{2} + 2 t + 2\right) x + 2$	$[1, 0]$	$[1]$	$z + (t^2 + t)$	$[1, 3]$
2.5.2.10a7.2	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(2 x^{4} + 2 x^{3} + 2 x^{2} + 2 x + 2\right) ( x^{5} + x^{2} + 1 ) + 4 x^{2} + 2$	$2$	$5$	$2$	$10$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$10$	$1$	$[2]$	$[1]$	$[2, 2, 2, 2]^{10}$	$[1,1,1,1]^{10}$	$[2,2,2]^{2}$	$[1,1,1]^{2}$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(2 t^{4} + 2 t\right) x + 4 t + 2$	$[1, 0]$	$[1]$	$z + (t^2 + t)$	$[1, 3]$
2.5.2.15a1.1	$10$	$( x^{5} + x^{2} + 1 )^{2} + 2$	$2$	$5$	$2$	$15$	$C_{10}$ (as 10T1)	$5$	$1$	$[3]$	$[2]$	$[3]^{5}$	$[2]^{5}$	$[\ ]$	$[\ ]$	$10$	$t^{5} + t^{2} + 1$	$x^{2} + 2$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.2	$10$	$( x^{5} + x^{2} + 1 )^{2} + 10$	$2$	$5$	$2$	$15$	$C_{10}$ (as 10T1)	$5$	$1$	$[3]$	$[2]$	$[3]^{5}$	$[2]^{5}$	$[\ ]$	$[\ ]$	$10$	$t^{5} + t^{2} + 1$	$x^{2} + 10$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.3	$10$	$( x^{5} + x^{2} + 1 )^{2} + 4 x^{3} ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$15$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[3]$	$[2]$	$[2, 2, 2, 2, 3]^{5}$	$[1,1,1,1,2]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(4 t^{3} + 4 t^{2} + 4 t\right) x + 2$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.4	$10$	$( x^{5} + x^{2} + 1 )^{2} + 4 x^{3} ( x^{5} + x^{2} + 1 ) + 10$	$2$	$5$	$2$	$15$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[3]$	$[2]$	$[2, 2, 2, 2, 3]^{5}$	$[1,1,1,1,2]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(4 t^{4} + 4 t + 4\right) x + 10$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.5	$10$	$( x^{5} + x^{2} + 1 )^{2} + 4 x ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$15$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[3]$	$[2]$	$[2, 2, 2, 2, 3]^{5}$	$[1,1,1,1,2]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(4 t^{4} + 4 t^{3} + 4 t + 4\right) x + 2$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.6	$10$	$( x^{5} + x^{2} + 1 )^{2} + 4 x ( x^{5} + x^{2} + 1 ) + 10$	$2$	$5$	$2$	$15$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[3]$	$[2]$	$[2, 2, 2, 2, 3]^{5}$	$[1,1,1,1,2]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(4 t^{4} + 4 t^{3} + 4 t + 4\right) x + 10$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.7	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(4 x^{2} + 4 x\right) ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$15$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[3]$	$[2]$	$[2, 2, 2, 2, 3]^{5}$	$[1,1,1,1,2]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(4 t^{4} + 4 t^{3} + 4\right) x + 2$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.8	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(4 x^{2} + 4 x\right) ( x^{5} + x^{2} + 1 ) + 10$	$2$	$5$	$2$	$15$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[3]$	$[2]$	$[2, 2, 2, 2, 3]^{5}$	$[1,1,1,1,2]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(4 t^{4} + 4 t^{3} + 4\right) x + 10$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.9	$10$	$( x^{5} + x^{2} + 1 )^{2} + 4 ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$15$	$C_{10}$ (as 10T1)	$5$	$1$	$[3]$	$[2]$	$[3]^{5}$	$[2]^{5}$	$[\ ]$	$[\ ]$	$10$	$t^{5} + t^{2} + 1$	$x^{2} + 4 x + 2$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.10	$10$	$( x^{5} + x^{2} + 1 )^{2} + 4 ( x^{5} + x^{2} + 1 ) + 10$	$2$	$5$	$2$	$15$	$C_{10}$ (as 10T1)	$5$	$1$	$[3]$	$[2]$	$[3]^{5}$	$[2]^{5}$	$[\ ]$	$[\ ]$	$10$	$t^{5} + t^{2} + 1$	$x^{2} + 4 x + 10$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.11	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(4 x^{3} + 4\right) ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$15$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[3]$	$[2]$	$[2, 2, 2, 2, 3]^{5}$	$[1,1,1,1,2]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(4 t^{4} + 4 t\right) x + 2$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.12	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(4 x^{3} + 4\right) ( x^{5} + x^{2} + 1 ) + 10$	$2$	$5$	$2$	$15$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[3]$	$[2]$	$[2, 2, 2, 2, 3]^{5}$	$[1,1,1,1,2]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(4 t^{4} + 4 t\right) x + 10$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.13	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(4 x + 4\right) ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$15$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[3]$	$[2]$	$[2, 2, 2, 2, 3]^{5}$	$[1,1,1,1,2]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(4 t^{4} + 4 t^{3} + 4 t\right) x + 2$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.14	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(4 x + 4\right) ( x^{5} + x^{2} + 1 ) + 10$	$2$	$5$	$2$	$15$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[3]$	$[2]$	$[2, 2, 2, 2, 3]^{5}$	$[1,1,1,1,2]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(4 t^{4} + 4 t^{3} + 4 t\right) x + 10$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.15	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(4 x^{2} + 4 x + 4\right) ( x^{5} + x^{2} + 1 ) + 2$	$2$	$5$	$2$	$15$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[3]$	$[2]$	$[2, 2, 2, 2, 3]^{5}$	$[1,1,1,1,2]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(4 t^{4} + 4 t^{2} + 4\right) x + 2$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$
2.5.2.15a1.16	$10$	$( x^{5} + x^{2} + 1 )^{2} + \left(4 x^{2} + 4 x + 4\right) ( x^{5} + x^{2} + 1 ) + 10$	$2$	$5$	$2$	$15$	$C_2 \times (C_2^4 : C_5)$ (as 10T14)	$5$	$1$	$[3]$	$[2]$	$[2, 2, 2, 2, 3]^{5}$	$[1,1,1,1,2]^{5}$	$[2,2,2,2]$	$[1,1,1,1]$	$2$	$t^{5} + t^{2} + 1$	$x^{2} + \left(4 t^{4} + 4 t^{2} + 4\right) x + 10$	$[2, 0]$	$[1]$	$z + (t^3 + t^2 + 1)$	$[1, 3]$

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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.