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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
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Results (16 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
2.4.4.44a1.5 $( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + 2$ $2$ $4$ $4$ $44$ $C_2^3.C_2^3$ (as 16T112) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.6 $( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $\OD_{16}:C_2^2$ (as 16T99) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.57 $( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_2^3.C_2^3$ (as 16T112) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + 8 x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.58 $( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $\OD_{16}.C_2^2$ (as 16T86) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + 8 x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.134 $( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $\OD_{32}:C_2$ (as 16T104) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8\right) x^{3} + 8 x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.138 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $\OD_{32}:C_2$ (as 16T104) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t + 8\right) x^{3} + 8 x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1151 $( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 2$ $2$ $4$ $4$ $44$ $C_2^3.C_2^3$ (as 16T112) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + 4 x^{2} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1152 $( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $\OD_{16}.C_2^2$ (as 16T86) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + 4 x^{2} + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1207 $( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_2^3.C_2^3$ (as 16T112) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + 4 x^{2} + 8 x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1208 $( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $\OD_{16}:C_2^2$ (as 16T99) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + 4 x^{2} + 8 x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1245 $( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 x^{2} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $\OD_{32}:C_2$ (as 16T104) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8\right) x^{3} + 4 x^{2} + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1246 $( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 x^{2} ( x^{4} + x + 1 ) + 24 x^{3} + 2$ $2$ $4$ $4$ $44$ $\OD_{32}:C_2$ (as 16T104) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8\right) x^{3} + 4 x^{2} + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1987 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_2^3.D_4$ (as 16T75) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2}\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1988 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_4^2:C_2^2$ (as 16T122) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2}\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1995 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 8 x + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_2^3.D_4$ (as 16T75) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1996 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 8 x + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_4^2:C_2^2$ (as 16T122) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 4]^{4}$ $[1,1,2,3]^{4}$ $[2,2]$ $[1,1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
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