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Properties

Label 2.1.20.40a
Base 2.1.1.0a1.1
Degree \(20\)
e \(20\)
f \(1\)
c \(40\)

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Defining polynomial

$x^{20} + 2 b_{18} x^{18} + 2 b_{14} x^{14} + 4 c_{28} x^{8} + 4 b_{27} x^{7} + 4 b_{26} x^{6} + 4 b_{25} x^{5} + 4 b_{23} x^{3} + 4 b_{22} x^{2} + 4 a_{21} x + 2$

Invariants

Residue field characteristic: $2$
Degree: $20$
Base field: $\Q_{2}$
Ramification index $e$: $20$
Residue field degree $f$: $1$
Discriminant exponent $c$: $40$
Artin slopes: $[\frac{12}{5},\frac{12}{5}]$
Swan slopes: $[\frac{7}{5},\frac{7}{5}]$
Means: $\langle\frac{7}{10},\frac{21}{20}\rangle$
Rams: $(7,7)$
Field count: $192$ (complete)
Ambiguity: $2$
Mass: $128$
Absolute Mass: $128$

Diagrams

Varying

Indices of inseparability: $[21,14,0]$ (show 64), $[21,18,0]$ (show 64), $[21,20,0]$ (show 64)
Associated inertia: $[4,2]$ (show 128), $[4,3]$ (show 64)
Jump Set: $[5,15,35]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing 1-50 of 64

Next   displayed columns for results
Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
2.1.20.40a1.65 $x^{20} + 2 x^{18} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.66 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.67 $x^{20} + 2 x^{18} + 4 x^{7} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.68 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{7} + 4 x + 2$ $C_2^6:F_5$ (as 20T194) $1280$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.69 $x^{20} + 2 x^{18} + 4 x^{6} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.70 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{6} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.71 $x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{6} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.72 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.73 $x^{20} + 2 x^{18} + 4 x^{5} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.74 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{5} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.75 $x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{5} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.76 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{7} + 4 x^{5} + 4 x + 2$ $C_2^6:F_5$ (as 20T194) $1280$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.77 $x^{20} + 2 x^{18} + 4 x^{6} + 4 x^{5} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.78 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{6} + 4 x^{5} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.79 $x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.80 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x + 2$ $C_2^6:F_5$ (as 20T194) $1280$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.81 $x^{20} + 2 x^{18} + 4 x^{3} + 4 x + 2$ $C_2^9.F_5$ (as 20T444) $10240$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.82 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.83 $x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{3} + 4 x + 2$ $C_2\wr F_5$ (as 20T129) $640$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.84 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{7} + 4 x^{3} + 4 x + 2$ $C_2\wr F_5$ (as 20T132) $640$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.85 $x^{20} + 2 x^{18} + 4 x^{6} + 4 x^{3} + 4 x + 2$ $C_2^9.F_5$ (as 20T444) $10240$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.86 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{6} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.87 $x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 4 x + 2$ $C_2\wr F_5$ (as 20T129) $640$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.88 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.89 $x^{20} + 2 x^{18} + 4 x^{5} + 4 x^{3} + 4 x + 2$ $C_2^9.F_5$ (as 20T444) $10240$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.90 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{5} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.91 $x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.92 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 4 x + 2$ $C_2\wr F_5$ (as 20T132) $640$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.93 $x^{20} + 2 x^{18} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 4 x + 2$ $C_2^9.F_5$ (as 20T444) $10240$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.94 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.95 $x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.96 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.97 $x^{20} + 2 x^{18} + 4 x^{2} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.98 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{2} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.99 $x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{2} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.100 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{7} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.101 $x^{20} + 2 x^{18} + 4 x^{6} + 4 x^{2} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.102 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{6} + 4 x^{2} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.103 $x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{6} + 4 x^{2} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.104 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{2} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.105 $x^{20} + 2 x^{18} + 4 x^{5} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.106 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{5} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.107 $x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{5} + 4 x^{2} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.108 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{2} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.109 $x^{20} + 2 x^{18} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.110 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.111 $x^{20} + 2 x^{18} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.112 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.113 $x^{20} + 2 x^{18} + 4 x^{3} + 4 x^{2} + 4 x + 2$ $C_2^9.F_5$ (as 20T444) $10240$ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.114 $x^{20} + 2 x^{18} + 4 x^{8} + 4 x^{3} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 18, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
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