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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 (32 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
2.4.4.44a1.231 $( x^{4} + x + 1 )^{4} + 4 x^{2} ( x^{4} + x + 1 )^{3} + 4 x^{3} ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T671) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.232 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2}\right) ( x^{4} + x + 1 )^{3} + 4 x^{3} ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T692) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t\right) x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.233 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 x^{3} ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T692) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.234 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 x^{3} ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T671) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.235 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 8\right) ( x^{4} + x + 1 )^{3} + 4 x^{3} ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T671) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t + 8\right) x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.236 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2} + 8\right) ( x^{4} + x + 1 )^{3} + 4 x^{3} ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T692) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t + 8\right) x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.237 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 8 x + 8\right) ( x^{4} + x + 1 )^{3} + 4 x^{3} ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T692) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 8 x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.238 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2} + 8 x + 8\right) ( x^{4} + x + 1 )^{3} + 4 x^{3} ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x^{2} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T671) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8\right) x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.807 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T663) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8\right) x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.808 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2} + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T663) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t + 8\right) x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.809 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 8 x + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T685) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t + 8\right) x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.810 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2} + 8 x + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T685) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 8 x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.811 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 12\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T663) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.812 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2} + 12\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T663) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.813 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 8 x + 12\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T685) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t\right) x^{3} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.814 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2} + 8 x + 12\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T685) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 4 t^{3} x^{2} + \left(8 t^{3} + 8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1411 $( x^{4} + x + 1 )^{4} + 4 x^{2} ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4\right) ( x^{4} + x + 1 )^{2} + 8 x^{3} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T671) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 8 x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1412 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2}\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4\right) ( x^{4} + x + 1 )^{2} + 8 x^{3} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T685) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8\right) x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1413 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4\right) ( x^{4} + x + 1 )^{2} + 8 x^{3} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T692) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t + 8\right) x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1414 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4\right) ( x^{4} + x + 1 )^{2} + 8 x^{3} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T663) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t + 8\right) x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1415 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 8\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4\right) ( x^{4} + x + 1 )^{2} + 8 x^{3} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T671) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1416 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2} + 8\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4\right) ( x^{4} + x + 1 )^{2} + 8 x^{3} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T685) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1417 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 8 x + 8\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4\right) ( x^{4} + x + 1 )^{2} + 8 x^{3} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T692) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1418 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2} + 8 x + 8\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4\right) ( x^{4} + x + 1 )^{2} + 8 x^{3} ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T663) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t\right) x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 24 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1667 $( x^{4} + x + 1 )^{4} + 4 x^{2} ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x^{2} + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T671) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t\right) x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1668 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2}\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x^{2} + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T663) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1669 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x^{2} + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T692) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8\right) x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1670 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x^{2} + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T685) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t + 8\right) x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1671 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 8\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x^{2} + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T671) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t + 8\right) x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1672 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2} + 8\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x^{2} + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T663) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 8 x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1673 $( x^{4} + x + 1 )^{4} + \left(4 x^{2} + 8 x + 8\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x^{2} + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2.C_4$ (as 16T692) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1674 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4 x^{2} + 8 x + 8\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{3} + 4 x^{2} + 4\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{3} + 8 x + 8\right) ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $D_4^2:C_4$ (as 16T685) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 2, 3, \frac{7}{2}, 4]^{4}$ $[1,1,1,2,\frac{5}{2},3]^{4}$ $[2,2,2,\frac{7}{2}]$ $[1,1,1,\frac{5}{2}]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + \left(4 t^{3} + 4\right) x^{2} + \left(8 t^{3} + 8 t^{2}\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
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