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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 (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
2.4.4.44a1.367 $( x^{4} + x + 1 )^{4} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $(C_4\times C_8).C_4$ (as 16T378) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + 8 t^{2} x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.368 $( x^{4} + x + 1 )^{4} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_4\times C_8):C_4$ (as 16T413) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + 8 t^{2} x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.369 $( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_4^2.(C_2\times C_4)$ (as 16T362) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8 t\right) x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.370 $( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_4^2.(C_2\times C_4)$ (as 16T362) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8 t\right) x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.371 $( x^{4} + x + 1 )^{4} + 8 x^{2} ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $(C_4\times C_8).C_4$ (as 16T378) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.372 $( x^{4} + x + 1 )^{4} + 8 x^{2} ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_4\times C_8):C_4$ (as 16T413) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.373 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2}\right) ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_4^2.(C_2\times C_4)$ (as 16T362) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2}\right) x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.374 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2}\right) ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_4^2.(C_2\times C_4)$ (as 16T362) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2}\right) x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.375 $( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $(C_4\times C_8).C_4$ (as 16T378) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.376 $( x^{4} + x + 1 )^{4} + 8 x ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_4\times C_8):C_4$ (as 16T413) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.377 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_4^2.(C_2\times C_4)$ (as 16T362) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t\right) x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.378 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_4^2.(C_2\times C_4)$ (as 16T362) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t\right) x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.379 $( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $(C_4\times C_8).C_4$ (as 16T378) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.380 $( x^{4} + x + 1 )^{4} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_4\times C_8):C_4$ (as 16T413) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.381 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $C_4^2.(C_2\times C_4)$ (as 16T362) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.382 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 8 x\right) ( x^{4} + x + 1 )^{3} + 4 x^{2} ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_4^2.(C_2\times C_4)$ (as 16T362) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + 4 t x^{2} + \left(8 t^{2} + 8\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.917 $( x^{4} + x + 1 )^{4} + 4 ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $(C_2\times C_4^2):C_4$ (as 16T314) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t + 8\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.918 $( x^{4} + x + 1 )^{4} + 4 ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_2\times C_4^2):C_4$ (as 16T314) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t + 8\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.923 $( x^{4} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $(C_2\times C_4^2):C_4$ (as 16T314) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + 8 x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.924 $( x^{4} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_2\times C_4^2):C_4$ (as 16T314) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{2} + 8 t\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.929 $( x^{4} + x + 1 )^{4} + 12 ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $(C_2\times C_4^2):C_4$ (as 16T314) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.930 $( x^{4} + x + 1 )^{4} + 12 ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_2\times C_4^2):C_4$ (as 16T314) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + 8 t x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.933 $( x^{4} + x + 1 )^{4} + \left(8 x + 12\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 2$ $2$ $4$ $4$ $44$ $(C_2\times C_4^2):C_4$ (as 16T314) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.934 $( x^{4} + x + 1 )^{4} + \left(8 x + 12\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + \left(8 x^{2} + 8 x\right) ( x^{4} + x + 1 ) + 16 x^{3} + 2$ $2$ $4$ $4$ $44$ $(C_2\times C_4^2):C_4$ (as 16T314) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 16 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.992 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_8.\OD_{16}$ (as 16T220) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8 t\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.994 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_8.\OD_{16}$ (as 16T220) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.996 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_8.\OD_{16}$ (as 16T220) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2}\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.998 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 8 x + 4\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_8.\OD_{16}$ (as 16T220) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + 8 t^{3} x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1000 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 12\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_8.\OD_{16}$ (as 16T220) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8 t + 8\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1002 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 12\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_8.\OD_{16}$ (as 16T220) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t + 8\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1004 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x + 12\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_8.\OD_{16}$ (as 16T220) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8 t^{2} + 8\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.4.4.44a1.1006 $( x^{4} + x + 1 )^{4} + \left(8 x^{3} + 8 x^{2} + 8 x + 12\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x\right) ( x^{4} + x + 1 )^{2} + 8 x ( x^{4} + x + 1 ) + 8 x^{3} + 2$ $2$ $4$ $4$ $44$ $C_8.\OD_{16}$ (as 16T220) $4$ $1$ $[3, 4]$ $[2,3]$ $[2, 2, 3, 3, 4]^{4}$ $[1,1,2,2,3]^{4}$ $[2,2,3]$ $[1,1,2]$ $t^{4} + t + 1$ $x^{4} + \left(8 t^{3} + 8\right) x^{3} + \left(4 t^{2} + 4 t\right) x^{2} + \left(8 t^{2} + 8 t\right) x + 8 t^{3} + 2$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
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