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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 (24 matches)

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Label 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 Unram. Ext. Eisen. Poly. Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
2.4.4.44a1.3 $( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 2$ $2$ $4$ $4$ $44$ $(C_2^2\times D_4):C_4$ (as 16T315) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t\right) x^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.4 $( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_2^2\times D_4):C_4$ (as 16T315) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t\right) x^{3} + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.55 $( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $(C_2^3\times C_4):C_4$ (as 16T292) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + 8 x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.56 $( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_2^3\times C_4):C_4$ (as 16T292) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + 8 x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.63 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $(C_2^3\times C_4):C_4$ (as 16T292) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8\right) x^{3} + 8 x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.64 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_2^3\times C_4):C_4$ (as 16T292) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8\right) x^{3} + 8 x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.133 $( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_2^4.C_8$ (as 16T306) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8\right) x^{3} + 8 x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.136 $( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_2^4.C_8$ (as 16T306) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t + 8\right) x^{3} + 8 x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.139 $( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_2^4.C_8$ (as 16T306) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t^{2} x^{3} + 8 x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.141 $( x^{4} + x + 1 )^{4} + \left(8 x + 8\right) ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_2^4.C_8$ (as 16T306) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + 8 x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1149 $( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 2$ $2$ $4$ $4$ $44$ $(C_2^3\times C_4):C_4$ (as 16T292) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + 4 x^{2} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1150 $( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_2^3\times C_4):C_4$ (as 16T292) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + 4 x^{2} + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1157 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 2$ $2$ $4$ $4$ $44$ $(C_2^3\times C_4):C_4$ (as 16T292) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8\right) x^{3} + 4 x^{2} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1158 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_2^3\times C_4):C_4$ (as 16T292) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8\right) x^{3} + 4 x^{2} + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1205 $( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $(C_2^2\times D_4):C_4$ (as 16T315) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + 4 x^{2} + 8 x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1206 $( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_2^2\times D_4):C_4$ (as 16T315) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + 4 x^{2} + 8 x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1243 $( x^{4} + x + 1 )^{4} + 4 ( x^{4} + x + 1 )^{2} + 8 x^{2} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_2^4.C_8$ (as 16T306) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8\right) x^{3} + 4 x^{2} + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1244 $( x^{4} + x + 1 )^{4} + 4 ( x^{4} + x + 1 )^{2} + 8 x^{2} ( x^{4} + x + 1 ) + 24 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_2^4.C_8$ (as 16T306) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8\right) x^{3} + 4 x^{2} + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1251 $( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 x^{2} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_2^4.C_8$ (as 16T306) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t^{2} x^{3} + 4 x^{2} + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1252 $( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 x^{2} ( x^{4} + x + 1 ) + 24 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_2^4.C_8$ (as 16T306) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t^{2} x^{3} + 4 x^{2} + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1985 $( x^{4} + x + 1 )^{4} + 4 ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_4^2.(C_2\times C_4)$ (as 16T285) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1986 $( x^{4} + x + 1 )^{4} + 4 ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_2^2\times D_4):C_4$ (as 16T263) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1991 $( x^{4} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_4^2.(C_2\times C_4)$ (as 16T285) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1992 $( x^{4} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_2^2\times D_4):C_4$ (as 16T263) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, 4]^{4}$ $[1,1,1,2,3]^{4}$ $[2,2,2]$ $[1,1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
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