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Properties

Label 2.1.2.3a1.2-1.6.17a
Base 2.1.2.3a1.2
Degree \(6\)
e \(6\)
f \(1\)
c \(17\)

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Defining polynomial

$x^{6} + \left(b_{23} \pi^{4} + b_{17} \pi^{3}\right) x^{5} + \left(b_{21} \pi^{4} + b_{15} \pi^{3}\right) x^{3} + \left(b_{19} \pi^{4} + b_{13} \pi^{3}\right) x + c_{24} \pi^{5} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $6$
Base field: $\Q_{2}(\sqrt{-2\cdot 5})$
Ramification index $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $17$
Absolute Artin slopes: $[3,4]$
Swan slopes: $[4]$
Means: $\langle2\rangle$
Rams: $(12)$
Field count: $64$ (complete)
Ambiguity: $2$
Mass: $64$
Absolute Mass: $32$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group: $D_{12}$ (show 2), $S_3\times D_4$ (show 2), $C_4:S_4$ (show 6), $D_4\times S_4$ (show 6), $C_2^4:D_{12}$ (show 24), $C_2\wr D_6$ (show 24)
Hidden Artin slopes: $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ (show 16), $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}$ (show 8), $[\frac{4}{3},\frac{4}{3},\frac{19}{6},\frac{19}{6}]^{2}$ (show 8), $[2]^{2}$ (show 2), $[\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ (show 16), $[\ ]^{2}$ (show 2), $[\frac{4}{3},\frac{4}{3},2]^{2}$ (show 2), $[2,\frac{8}{3},\frac{8}{3}]^{2}$ (show 4), $[\frac{8}{3},\frac{8}{3}]^{2}$ (show 4), $[\frac{4}{3},\frac{4}{3}]^{2}$ (show 2)
Indices of inseparability: $[24,12,0]$
Associated inertia: $[2,1,1]$
Jump Set: $[3,9,21]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 24

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Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
2.1.12.35a1.73 $x^{12} + 8 x^{5} + 10$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.74 $x^{12} + 8 x^{5} + 26$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.75 $x^{12} + 8 x^{11} + 8 x^{5} + 10$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.76 $x^{12} + 8 x^{11} + 8 x^{5} + 26$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.77 $x^{12} + 8 x^{7} + 8 x^{5} + 10$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.78 $x^{12} + 8 x^{7} + 8 x^{5} + 26$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.79 $x^{12} + 8 x^{11} + 8 x^{7} + 8 x^{5} + 10$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.80 $x^{12} + 8 x^{11} + 8 x^{7} + 8 x^{5} + 26$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[\frac{4}{3}, \frac{4}{3}, 2, 3, \frac{19}{6}, \frac{19}{6}, 4]_{3}^{2}$ $[\frac{1}{3},\frac{1}{3},1,2,\frac{13}{6},\frac{13}{6},3]_{3}^{2}$ $[\frac{4}{3},\frac{4}{3},2,\frac{19}{6},\frac{19}{6}]^{2}$ $[\frac{1}{3},\frac{1}{3},1,\frac{13}{6},\frac{13}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.113 $x^{12} + 8 x^{3} + 8 x + 10$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.114 $x^{12} + 8 x^{3} + 8 x + 26$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.115 $x^{12} + 8 x^{11} + 8 x^{3} + 8 x + 10$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.116 $x^{12} + 8 x^{11} + 8 x^{3} + 8 x + 26$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.117 $x^{12} + 8 x^{9} + 8 x^{3} + 8 x + 10$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.118 $x^{12} + 8 x^{9} + 8 x^{3} + 8 x + 26$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.119 $x^{12} + 8 x^{11} + 8 x^{9} + 8 x^{3} + 8 x + 10$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.120 $x^{12} + 8 x^{11} + 8 x^{9} + 8 x^{3} + 8 x + 26$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.121 $x^{12} + 8 x^{5} + 8 x^{3} + 8 x + 10$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.122 $x^{12} + 8 x^{5} + 8 x^{3} + 8 x + 26$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.123 $x^{12} + 8 x^{11} + 8 x^{5} + 8 x^{3} + 8 x + 10$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.124 $x^{12} + 8 x^{11} + 8 x^{5} + 8 x^{3} + 8 x + 26$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.125 $x^{12} + 8 x^{9} + 8 x^{5} + 8 x^{3} + 8 x + 10$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.126 $x^{12} + 8 x^{9} + 8 x^{5} + 8 x^{3} + 8 x + 26$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.127 $x^{12} + 8 x^{11} + 8 x^{9} + 8 x^{5} + 8 x^{3} + 8 x + 10$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
2.1.12.35a1.128 $x^{12} + 8 x^{11} + 8 x^{9} + 8 x^{5} + 8 x^{3} + 8 x + 26$ $C_2\wr D_6$ (as 12T193) $768$ $2$ $[2, \frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4]_{3}^{2}$ $[1,\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3]_{3}^{2}$ $[2,\frac{8}{3},\frac{8}{3},\frac{23}{6},\frac{23}{6}]^{2}$ $[1,\frac{5}{3},\frac{5}{3},\frac{17}{6},\frac{17}{6}]^{2}$ $[24, 12, 0]$ $[2, 1, 1]$ $z^8 + z^4 + 1,z^2 + 1,z + 1$ $[3, 9, 21]$
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