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Properties

Label 2.1.10.19a1.39-1.2.16a
Base 2.1.10.19a1.39
Degree \(2\)
e \(2\)
f \(1\)
c \(16\)

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Defining polynomial

$x^{2} + \left(b_{29} \pi^{15} + b_{27} \pi^{14} + b_{25} \pi^{13} + b_{23} \pi^{12} + b_{21} \pi^{11} + b_{19} \pi^{10} + b_{17} \pi^{9} + a_{15} \pi^{8}\right) x + c_{30} \pi^{16} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.1.10.19a1.39
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
Absolute Artin slopes: $[3,\frac{7}{2}]$
Swan slopes: $[15]$
Means: $\langle\frac{15}{2}\rangle$
Rams: $(15)$
Field count: $128$ (complete)
Ambiguity: $2$
Mass: $128$
Absolute Mass: $64$

Diagrams

Varying

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

Indices of inseparability: $[35,20,0]$
Associated inertia: $[4,1,1]$
Jump Set: $[5,15,35]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing 1-50 of 128

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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
2.1.20.54b1.5121 $x^{20} + 4 x^{15} + 4 x^{10} + 4 x^{2} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5122 $x^{20} + 4 x^{15} + 12 x^{10} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5123 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{9} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5124 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{9} + 4 x^{2} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5125 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{7} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5126 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{7} + 4 x^{2} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5127 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5128 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{9} + 8 x^{7} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5129 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{3} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5130 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{3} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5131 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{9} + 8 x^{3} + 4 x^{2} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5132 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{9} + 8 x^{3} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5133 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{7} + 8 x^{3} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5134 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{7} + 8 x^{3} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5135 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{3} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5136 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{3} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5137 $x^{20} + 4 x^{15} + 4 x^{10} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5138 $x^{20} + 4 x^{15} + 12 x^{10} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5139 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{9} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5140 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{9} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5141 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{7} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5142 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{7} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5143 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 4 x^{2} + 8 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5144 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{9} + 8 x^{7} + 4 x^{2} + 8 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5145 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{3} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5146 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{3} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5147 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{9} + 8 x^{3} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5148 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{9} + 8 x^{3} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5149 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{7} + 8 x^{3} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5150 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{7} + 8 x^{3} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5151 $x^{20} + 4 x^{15} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{3} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5152 $x^{20} + 4 x^{15} + 12 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{3} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5313 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 4 x^{10} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5314 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 12 x^{10} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5315 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 4 x^{10} + 8 x^{9} + 4 x^{2} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5316 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 12 x^{10} + 8 x^{9} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5317 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 4 x^{10} + 8 x^{7} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5318 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 12 x^{10} + 8 x^{7} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5319 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5320 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 12 x^{10} + 8 x^{9} + 8 x^{7} + 4 x^{2} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5321 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 4 x^{10} + 8 x^{3} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5322 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 12 x^{10} + 8 x^{3} + 4 x^{2} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5323 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 4 x^{10} + 8 x^{9} + 8 x^{3} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5324 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 12 x^{10} + 8 x^{9} + 8 x^{3} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5325 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 4 x^{10} + 8 x^{7} + 8 x^{3} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5326 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 12 x^{10} + 8 x^{7} + 8 x^{3} + 4 x^{2} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5327 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 4 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{3} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5328 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 12 x^{10} + 8 x^{9} + 8 x^{7} + 8 x^{3} + 4 x^{2} + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5329 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 4 x^{10} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.54b1.5330 $x^{20} + 4 x^{19} + 4 x^{18} + 4 x^{15} + 12 x^{10} + 4 x^{2} + 8 x + 2$ $C_2^{10}.C_2\wr F_5$ (as 20T946) $655360$ $2$ not computed not computed not computed not computed $[35, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
Next   displayed columns for results