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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 (20 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
2.4.4.44a1.1 $( x^{4} + x + 1 )^{4} + 2$ $2$ $4$ $4$ $44$ $C_4 \times D_4$ (as 16T19) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.2 $( x^{4} + x + 1 )^{4} + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_8:C_2):C_2$ (as 16T16) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.7 $( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 2$ $2$ $4$ $4$ $44$ $C_4 \times D_4$ (as 16T19) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.8 $( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_8:C_2):C_2$ (as 16T16) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.59 $( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_4^2:C_2$ (as 16T17) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t + 8\right) x^{3} + 8 x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.60 $( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_2 \times (C_8:C_2)$ (as 16T15) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + 8 x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.135 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2}\right) ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_{16} : C_2$ (as 16T22) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8\right) x^{3} + 8 x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.140 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 8\right) ( x^{4} + x + 1 )^{3} + \left(8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_{16} : C_2$ (as 16T22) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + 8 x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1153 $( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 2$ $2$ $4$ $4$ $44$ $C_4^2:C_2$ (as 16T17) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + 4 x^{2} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1154 $( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_2 \times (C_8:C_2)$ (as 16T15) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + 4 x^{2} + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1203 $( x^{4} + x + 1 )^{4} + 4 ( x^{4} + x + 1 )^{2} + 8 ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_4 \times D_4$ (as 16T19) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + 4 x^{2} + 8 x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1204 $( x^{4} + x + 1 )^{4} + 4 ( x^{4} + x + 1 )^{2} + 8 ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_8:C_2):C_2$ (as 16T16) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + 4 x^{2} + 8 x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1209 $( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_4 \times D_4$ (as 16T19) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + 4 x^{2} + 8 x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1210 $( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_8:C_2):C_2$ (as 16T16) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + 4 x^{2} + 8 x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1247 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2}\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 x^{2} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_{16} : C_2$ (as 16T22) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8\right) x^{3} + 4 x^{2} + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1248 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2}\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 x^{2} ( x^{4} + x + 1 ) + 24 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_{16} : C_2$ (as 16T22) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8\right) x^{3} + 4 x^{2} + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1989 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_4\wr C_2$ (as 16T28) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8 t\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1990 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4:C_4$ (as 16T26) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8 t\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1993 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_4\wr C_2$ (as 16T28) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1994 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4:C_4$ (as 16T26) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 3, 4]^{4}$ $[1,2,3]^{4}$ $[2]$ $[1]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
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