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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 (12 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
2.4.4.44a1.53 $( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_4:C_4$ (as 16T8) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + 8 x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.54 $( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_8: C_2$ (as 16T6) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + 8 x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.61 $( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_4:C_4$ (as 16T8) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + 8 x^{3} + 8 x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.62 $( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 )^{3} + 8 ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_8: C_2$ (as 16T6) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + 8 x^{3} + 8 x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1147 $( x^{4} + x + 1 )^{4} + 4 ( x^{4} + x + 1 )^{2} + 2$ $2$ $4$ $4$ $44$ $C_4^2$ (as 16T4) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + 4 x^{2} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1148 $( x^{4} + x + 1 )^{4} + 4 ( x^{4} + x + 1 )^{2} + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_8\times C_2$ (as 16T5) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + 4 x^{2} + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1155 $( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 2$ $2$ $4$ $4$ $44$ $C_4^2$ (as 16T4) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + 8 x^{3} + 4 x^{2} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1156 $( x^{4} + x + 1 )^{4} + 8 ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_8\times C_2$ (as 16T5) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + 8 x^{3} + 4 x^{2} + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1249 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 x^{2} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_{16}$ (as 16T1) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8 t + 8\right) x^{3} + 4 x^{2} + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1250 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 x^{2} ( x^{4} + x + 1 ) + 24 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_{16}$ (as 16T1) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8 t + 8\right) x^{3} + 4 x^{2} + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1253 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x + 8\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 x^{2} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_{16}$ (as 16T1) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8 t\right) x^{3} + 4 x^{2} + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1254 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x + 8\right) ( x^{4} + x + 1 )^{3} + 4 ( x^{4} + x + 1 )^{2} + 8 x^{2} ( x^{4} + x + 1 ) + 24 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_{16}$ (as 16T1) $4$ $1$ $[3, 4]$ $[2,3]$ $[3, 4]^{4}$ $[2,3]^{4}$ $[\ ]$ $[\ ]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8 t\right) x^{3} + 4 x^{2} + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
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