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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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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
2.1.8.12b1.3 $x^{8} + 2 x^{7} + 2 x^{5} + 2 x^{2} + 2$ $2$ $1$ $8$ $12$ $S_4\times C_2$ (as 8T24) $2$ $3$ $[\frac{4}{3}, \frac{4}{3}, 2]$ $[\frac{1}{3},\frac{1}{3},1]$ $[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $t + 1$ $x^{8} + 2 x^{7} + 2 x^{5} + 2 x^{2} + 2$ $[5, 2, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 5, 10]$
2.1.8.12b1.4 $x^{8} + 2 x^{7} + 2 x^{5} + 2 x^{2} + 6$ $2$ $1$ $8$ $12$ $S_4\times C_2$ (as 8T24) $2$ $3$ $[\frac{4}{3}, \frac{4}{3}, 2]$ $[\frac{1}{3},\frac{1}{3},1]$ $[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $t + 1$ $x^{8} + 2 x^{7} + 2 x^{5} + 2 x^{2} + 6$ $[5, 2, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 5, 16]$
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