↑

Properties

Label 2.1.20.59a
Base 2.1.1.0a1.1
Degree \(20\)
e \(20\)
f \(1\)
c \(59\)

Related objects

Downloads

Learn more

Defining polynomial

$x^{20} + 8 b_{59} x^{19} + 4 b_{38} x^{18} + 8 b_{57} x^{17} + 8 b_{55} x^{15} + 4 b_{34} x^{14} + 8 b_{53} x^{13} + 8 b_{51} x^{11} + 4 b_{30} x^{10} + 8 b_{49} x^{9} + 8 b_{47} x^{7} + 4 b_{26} x^{6} + 8 b_{45} x^{5} + 8 b_{43} x^{3} + 4 b_{22} x^{2} + 8 b_{41} x + 8 c_{40} + 16 c_{60} + 2$

Invariants

Residue field characteristic: $2$
Degree: $20$
Base field: $\Q_{2}$
Ramification index $e$: $20$
Residue field degree $f$: $1$
Discriminant exponent $c$: $59$
Artin slopes: $[3,4]$
Swan slopes: $[2,3]$
Means: $\langle1,2\rangle$
Rams: $(10,20)$
Field count: $66560$ (complete)
Ambiguity: $4$
Mass: $32768$
Absolute Mass: $32768$

Diagrams

Varying

Indices of inseparability: $[40,20,0]$
Associated inertia: $[4,1,1]$
Jump Set: $[5,15,35]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
Sort order Select

Showing 1-50 of at least 1000

Next   To download results, determine the number of 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
2.1.20.59a1.1 $x^{20} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.2 $x^{20} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.3 $x^{20} + 8 x^{19} + 2$ $D_4\times C_2^4:F_5$ (as 20T261) $2560$ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.4 $x^{20} + 8 x^{19} + 18$ $D_4\times C_2^4:F_5$ (as 20T261) $2560$ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.5 $x^{20} + 8 x^{17} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.6 $x^{20} + 8 x^{17} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.7 $x^{20} + 8 x^{19} + 8 x^{17} + 2$ $D_4\times C_2^4:F_5$ (as 20T261) $2560$ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.8 $x^{20} + 8 x^{19} + 8 x^{17} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.9 $x^{20} + 8 x^{13} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.10 $x^{20} + 8 x^{13} + 18$ $D_4\times C_2^4:F_5$ (as 20T261) $2560$ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.11 $x^{20} + 8 x^{19} + 8 x^{13} + 2$ $D_4\times C_2^4:F_5$ (as 20T261) $2560$ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.12 $x^{20} + 8 x^{19} + 8 x^{13} + 18$ $D_4\times C_2^4:F_5$ (as 20T261) $2560$ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.13 $x^{20} + 8 x^{17} + 8 x^{13} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.14 $x^{20} + 8 x^{17} + 8 x^{13} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.15 $x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{13} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.16 $x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{13} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.17 $x^{20} + 8 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.18 $x^{20} + 8 x^{11} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.19 $x^{20} + 8 x^{19} + 8 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.20 $x^{20} + 8 x^{19} + 8 x^{11} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.21 $x^{20} + 8 x^{17} + 8 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.22 $x^{20} + 8 x^{17} + 8 x^{11} + 18$ $D_4\times C_2^4:F_5$ (as 20T261) $2560$ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.23 $x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.24 $x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{11} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.25 $x^{20} + 8 x^{13} + 8 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.26 $x^{20} + 8 x^{13} + 8 x^{11} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.27 $x^{20} + 8 x^{19} + 8 x^{13} + 8 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.28 $x^{20} + 8 x^{19} + 8 x^{13} + 8 x^{11} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.29 $x^{20} + 8 x^{17} + 8 x^{13} + 8 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.30 $x^{20} + 8 x^{17} + 8 x^{13} + 8 x^{11} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.31 $x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{13} + 8 x^{11} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.32 $x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{13} + 8 x^{11} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.33 $x^{20} + 8 x^{9} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.34 $x^{20} + 8 x^{9} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.35 $x^{20} + 8 x^{19} + 8 x^{9} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.36 $x^{20} + 8 x^{19} + 8 x^{9} + 18$ $C_2^8.(D_4\times F_5)$ (as 20T633) $40960$ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.37 $x^{20} + 8 x^{17} + 8 x^{9} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.38 $x^{20} + 8 x^{17} + 8 x^{9} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.39 $x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{9} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.40 $x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{9} + 18$ $C_2^8.(D_4\times F_5)$ (as 20T633) $40960$ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.41 $x^{20} + 8 x^{13} + 8 x^{9} + 2$ $C_2^8.(D_4\times F_5)$ (as 20T633) $40960$ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.42 $x^{20} + 8 x^{13} + 8 x^{9} + 18$ $C_2^8.(D_4\times F_5)$ (as 20T633) $40960$ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.43 $x^{20} + 8 x^{19} + 8 x^{13} + 8 x^{9} + 2$ $C_2^8.(D_4\times F_5)$ (as 20T633) $40960$ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.44 $x^{20} + 8 x^{19} + 8 x^{13} + 8 x^{9} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.45 $x^{20} + 8 x^{17} + 8 x^{13} + 8 x^{9} + 2$ $C_2^8.(D_4\times F_5)$ (as 20T633) $40960$ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.46 $x^{20} + 8 x^{17} + 8 x^{13} + 8 x^{9} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.47 $x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{13} + 8 x^{9} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.48 $x^{20} + 8 x^{19} + 8 x^{17} + 8 x^{13} + 8 x^{9} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.49 $x^{20} + 8 x^{11} + 8 x^{9} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
2.1.20.59a1.50 $x^{20} + 8 x^{11} + 8 x^{9} + 18$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[40, 20, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 15, 35]$
Next   To download results, determine the number of results.