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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 (16 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
2.1.16.50h2.9	$16$	$x^{16} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 2$	$2$	$1$	$16$	$50$	$\SL(2,3):C_2$ (as 16T60)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 2$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.10	$16$	$x^{16} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 18$	$2$	$1$	$16$	$50$	$\SL(2,3):C_2$ (as 16T60)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 18$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.11	$16$	$x^{16} + 8 x^{15} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 2$	$2$	$1$	$16$	$50$	$\SL(2,3):C_2$ (as 16T60)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 8 x^{15} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 2$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.12	$16$	$x^{16} + 8 x^{15} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 18$	$2$	$1$	$16$	$50$	$\SL(2,3):C_2$ (as 16T60)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 8 x^{15} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 18$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.89	$16$	$x^{16} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 10$	$2$	$1$	$16$	$50$	$\SL(2,3):C_2$ (as 16T60)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 10$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.90	$16$	$x^{16} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 26$	$2$	$1$	$16$	$50$	$\SL(2,3):C_2$ (as 16T60)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 26$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.91	$16$	$x^{16} + 8 x^{15} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 10$	$2$	$1$	$16$	$50$	$\SL(2,3):C_2$ (as 16T60)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 8 x^{15} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 10$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.92	$16$	$x^{16} + 8 x^{15} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 26$	$2$	$1$	$16$	$50$	$\SL(2,3):C_2$ (as 16T60)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 8 x^{15} + 2 x^{12} + 8 x^{9} + 2 x^{8} + 4 x^{6} + 8 x^{3} + 26$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.193	$16$	$x^{16} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 2$	$2$	$1$	$16$	$50$	$C_4\times A_4$ (as 16T57)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 2$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.194	$16$	$x^{16} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 18$	$2$	$1$	$16$	$50$	$C_4\times A_4$ (as 16T57)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 18$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.195	$16$	$x^{16} + 8 x^{15} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 2$	$2$	$1$	$16$	$50$	$C_4\times A_4$ (as 16T57)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 8 x^{15} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 2$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.196	$16$	$x^{16} + 8 x^{15} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 18$	$2$	$1$	$16$	$50$	$C_4\times A_4$ (as 16T57)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 8 x^{15} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 18$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.273	$16$	$x^{16} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 10$	$2$	$1$	$16$	$50$	$C_4\times A_4$ (as 16T57)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 10$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.274	$16$	$x^{16} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 26$	$2$	$1$	$16$	$50$	$C_4\times A_4$ (as 16T57)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 26$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.275	$16$	$x^{16} + 8 x^{15} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 10$	$2$	$1$	$16$	$50$	$C_4\times A_4$ (as 16T57)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 8 x^{15} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 10$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$
2.1.16.50h2.276	$16$	$x^{16} + 8 x^{15} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 26$	$2$	$1$	$16$	$50$	$C_4\times A_4$ (as 16T57)	$3$	$1$	$[2, 2, 3, 4]$	$[1,1,2,3]$	$[2, 2, 3, 4]^{3}$	$[1,1,2,3]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$4$	$t + 1$	$x^{16} + 8 x^{15} + 4 x^{14} + 2 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 26$	$[35, 22, 12, 8, 0]$	$[3, 1, 1]$	$z^{12} + z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 7, 15, 31]$

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