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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 (18 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
2.1.12.18b1.4 $x^{12} + 2 x^{9} + 2 x^{7} + 2 x^{2} + 2$ $2$ $1$ $12$ $18$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{4}{3}, 2]$ $[\frac{1}{3},1]$ $[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1]_{3}^{2}$ $[\frac{4}{3}]^{2}$ $[\frac{1}{3}]^{2}$ $t + 1$ $x^{12} + 2 x^{9} + 2 x^{7} + 2 x^{2} + 2$ $[7, 2, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 7, 14]$
2.1.12.18b1.5 $x^{12} + 2 x^{9} + 2 x^{7} + 2 x^{2} + 6$ $2$ $1$ $12$ $18$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{4}{3}, 2]$ $[\frac{1}{3},1]$ $[\frac{4}{3}, \frac{4}{3}, 2]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1]_{3}^{2}$ $[\frac{4}{3}]^{2}$ $[\frac{1}{3}]^{2}$ $t + 1$ $x^{12} + 2 x^{9} + 2 x^{7} + 2 x^{2} + 6$ $[7, 2, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 7, 24]$
2.1.12.24b1.1 $x^{12} + 2 x^{2} + 4 x + 2$ $2$ $1$ $12$ $24$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{4}{3}, 3]$ $[\frac{1}{3},2]$ $[\frac{4}{3}, \frac{4}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},2]_{3}^{2}$ $[\frac{4}{3}]^{2}$ $[\frac{1}{3}]^{2}$ $t + 1$ $x^{12} + 2 x^{2} + 4 x + 2$ $[13, 2, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 7, 19]$
2.1.12.24b1.2 $x^{12} + 2 x^{2} + 4 x + 10$ $2$ $1$ $12$ $24$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{4}{3}, 3]$ $[\frac{1}{3},2]$ $[\frac{4}{3}, \frac{4}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},2]_{3}^{2}$ $[\frac{4}{3}]^{2}$ $[\frac{1}{3}]^{2}$ $t + 1$ $x^{12} + 2 x^{2} + 4 x + 10$ $[13, 2, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 7, 19]$
2.1.12.24b1.10 $x^{12} + 4 x^{9} + 4 x^{7} + 2 x^{2} + 4 x + 2$ $2$ $1$ $12$ $24$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{4}{3}, 3]$ $[\frac{1}{3},2]$ $[\frac{4}{3}, \frac{4}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},2]_{3}^{2}$ $[\frac{4}{3}]^{2}$ $[\frac{1}{3}]^{2}$ $t + 1$ $x^{12} + 4 x^{9} + 4 x^{7} + 2 x^{2} + 4 x + 2$ $[13, 2, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 7, 19]$
2.1.12.24b1.11 $x^{12} + 4 x^{9} + 4 x^{7} + 2 x^{2} + 4 x + 10$ $2$ $1$ $12$ $24$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{4}{3}, 3]$ $[\frac{1}{3},2]$ $[\frac{4}{3}, \frac{4}{3}, 3]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},2]_{3}^{2}$ $[\frac{4}{3}]^{2}$ $[\frac{1}{3}]^{2}$ $t + 1$ $x^{12} + 4 x^{9} + 4 x^{7} + 2 x^{2} + 4 x + 10$ $[13, 2, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 7, 19]$
2.1.12.24c1.4 $x^{12} + 4 x^{8} + 2 x^{6} + 4 x^{5} + 4 x + 2$ $2$ $1$ $12$ $24$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[2, \frac{8}{3}]$ $[1,\frac{5}{3}]$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $t + 1$ $x^{12} + 4 x^{8} + 2 x^{6} + 4 x^{5} + 4 x + 2$ $[13, 6, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 13, 25]$
2.1.12.24c1.6 $x^{12} + 4 x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{5} + 4 x + 2$ $2$ $1$ $12$ $24$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[2, \frac{8}{3}]$ $[1,\frac{5}{3}]$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $t + 1$ $x^{12} + 4 x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{5} + 4 x + 2$ $[13, 6, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 13, 25]$
2.1.12.24c1.14 $x^{12} + 4 x^{8} + 2 x^{6} + 4 x^{5} + 4 x + 6$ $2$ $1$ $12$ $24$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[2, \frac{8}{3}]$ $[1,\frac{5}{3}]$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $t + 1$ $x^{12} + 4 x^{8} + 2 x^{6} + 4 x^{5} + 4 x + 6$ $[13, 6, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 6, 24]$
2.1.12.24c1.16 $x^{12} + 4 x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{5} + 4 x + 6$ $2$ $1$ $12$ $24$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[2, \frac{8}{3}]$ $[1,\frac{5}{3}]$ $[2, \frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $t + 1$ $x^{12} + 4 x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{5} + 4 x + 6$ $[13, 6, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 6, 24]$
2.1.12.28c1.1 $x^{12} + 2 x^{10} + 4 x^{5} + 2$ $2$ $1$ $12$ $28$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{8}{3}, 3]$ $[\frac{5}{3},2]$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $t + 1$ $x^{12} + 2 x^{10} + 4 x^{5} + 2$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.2 $x^{12} + 2 x^{10} + 4 x^{5} + 10$ $2$ $1$ $12$ $28$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{8}{3}, 3]$ $[\frac{5}{3},2]$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $t + 1$ $x^{12} + 2 x^{10} + 4 x^{5} + 10$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.7 $x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 4 x^{5} + 2$ $2$ $1$ $12$ $28$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{8}{3}, 3]$ $[\frac{5}{3},2]$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $t + 1$ $x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 4 x^{5} + 2$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.8 $x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 4 x^{5} + 10$ $2$ $1$ $12$ $28$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{8}{3}, 3]$ $[\frac{5}{3},2]$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $t + 1$ $x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 4 x^{5} + 10$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.27 $x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{6} + 4 x^{5} + 2$ $2$ $1$ $12$ $28$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{8}{3}, 3]$ $[\frac{5}{3},2]$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $t + 1$ $x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{6} + 4 x^{5} + 2$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.28 $x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{6} + 4 x^{5} + 10$ $2$ $1$ $12$ $28$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{8}{3}, 3]$ $[\frac{5}{3},2]$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $t + 1$ $x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{6} + 4 x^{5} + 10$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.29 $x^{12} + 2 x^{10} + 4 x^{9} + 4 x^{6} + 4 x^{5} + 2$ $2$ $1$ $12$ $28$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{8}{3}, 3]$ $[\frac{5}{3},2]$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $t + 1$ $x^{12} + 2 x^{10} + 4 x^{9} + 4 x^{6} + 4 x^{5} + 2$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.28c1.30 $x^{12} + 2 x^{10} + 4 x^{9} + 4 x^{6} + 4 x^{5} + 10$ $2$ $1$ $12$ $28$ $C_2 \times S_4$ (as 12T24) $2$ $3$ $[\frac{8}{3}, 3]$ $[\frac{5}{3},2]$ $[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$ $[\frac{8}{3}]^{2}$ $[\frac{5}{3}]^{2}$ $t + 1$ $x^{12} + 2 x^{10} + 4 x^{9} + 4 x^{6} + 4 x^{5} + 10$ $[17, 10, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
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