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Degree $n$ Residue characteristic $p$ Discriminant exponent $c$ Galois group
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Results (4 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
2.2.8.40c1.9 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $2$ $2$ $8$ $40$ $C_2^3:S_4$ (as 16T429) $2$ $3$ $[\frac{8}{3}, \frac{8}{3}, 3]$ $[\frac{5}{3},\frac{5}{3},2]$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3}]_{3}$ $t^{2} + t + 1$ $x^{8} + \left(4 t + 4\right) x^{7} + 4 x^{5} + 4 t x^{4} + 4 x^{2} + 2$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.10 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ $2$ $2$ $8$ $40$ $C_2^3.S_4$ (as 16T434) $2$ $3$ $[\frac{8}{3}, \frac{8}{3}, 3]$ $[\frac{5}{3},\frac{5}{3},2]$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3}]_{3}$ $t^{2} + t + 1$ $x^{8} + \left(4 t + 4\right) x^{7} + 4 x^{5} + 4 t x^{4} + 4 x^{2} + 8 t + 2$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.13 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$ $2$ $2$ $8$ $40$ $C_2^3:S_4$ (as 16T429) $2$ $3$ $[\frac{8}{3}, \frac{8}{3}, 3]$ $[\frac{5}{3},\frac{5}{3},2]$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3}]_{3}$ $t^{2} + t + 1$ $x^{8} + 4 t x^{7} + 4 x^{5} + 4 t x^{4} + 4 x^{2} + 2$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.40c1.14 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 8 x + 2$ $2$ $2$ $8$ $40$ $C_2^3.S_4$ (as 16T434) $2$ $3$ $[\frac{8}{3}, \frac{8}{3}, 3]$ $[\frac{5}{3},\frac{5}{3},2]$ $[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3}]_{3}$ $[\frac{1}{3},\frac{1}{3}]_{3}$ $t^{2} + t + 1$ $x^{8} + 4 t x^{7} + 4 x^{5} + 4 t x^{4} + 4 x^{2} + 8 t + 2$ $[13, 10, 8, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7, 15]$
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