↑

Properties

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

Related objects

Downloads

Learn more

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
Sort order Select

Showing 1-50 of 192

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
2.1.20.40a1.1 $x^{20} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.2 $x^{20} + 4 x^{8} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.3 $x^{20} + 4 x^{7} + 4 x + 2$ $C_2^8.F_5$ (as 20T317) $5120$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.4 $x^{20} + 4 x^{8} + 4 x^{7} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.5 $x^{20} + 4 x^{6} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.6 $x^{20} + 4 x^{8} + 4 x^{6} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.7 $x^{20} + 4 x^{7} + 4 x^{6} + 4 x + 2$ $C_2^8.F_5$ (as 20T317) $5120$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.8 $x^{20} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.9 $x^{20} + 4 x^{5} + 4 x + 2$ $C_2^4:F_5$ (as 20T80) $320$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.10 $x^{20} + 4 x^{8} + 4 x^{5} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.11 $x^{20} + 4 x^{7} + 4 x^{5} + 4 x + 2$ $C_2^8.F_5$ (as 20T317) $5120$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.12 $x^{20} + 4 x^{8} + 4 x^{7} + 4 x^{5} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.13 $x^{20} + 4 x^{6} + 4 x^{5} + 4 x + 2$ $C_2^4:F_5$ (as 20T80) $320$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.14 $x^{20} + 4 x^{8} + 4 x^{6} + 4 x^{5} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.15 $x^{20} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x + 2$ $C_2^8.F_5$ (as 20T317) $5120$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.16 $x^{20} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.17 $x^{20} + 4 x^{3} + 4 x + 2$ $C_2^6:F_5$ (as 20T194) $1280$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.18 $x^{20} + 4 x^{8} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.19 $x^{20} + 4 x^{7} + 4 x^{3} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.20 $x^{20} + 4 x^{8} + 4 x^{7} + 4 x^{3} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.21 $x^{20} + 4 x^{6} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.22 $x^{20} + 4 x^{8} + 4 x^{6} + 4 x^{3} + 4 x + 2$ $C_2^6:F_5$ (as 20T194) $1280$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.23 $x^{20} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.24 $x^{20} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.25 $x^{20} + 4 x^{5} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.26 $x^{20} + 4 x^{8} + 4 x^{5} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.27 $x^{20} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.28 $x^{20} + 4 x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.29 $x^{20} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.30 $x^{20} + 4 x^{8} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 4 x + 2$ $C_2^6:F_5$ (as 20T194) $1280$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.31 $x^{20} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.32 $x^{20} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{3} + 4 x + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.33 $x^{20} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.34 $x^{20} + 4 x^{8} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.35 $x^{20} + 4 x^{7} + 4 x^{2} + 4 x + 2$ $C_2^8.F_5$ (as 20T317) $5120$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.36 $x^{20} + 4 x^{8} + 4 x^{7} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.37 $x^{20} + 4 x^{6} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.38 $x^{20} + 4 x^{8} + 4 x^{6} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.39 $x^{20} + 4 x^{7} + 4 x^{6} + 4 x^{2} + 4 x + 2$ $C_2^8.F_5$ (as 20T317) $5120$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.40 $x^{20} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.41 $x^{20} + 4 x^{5} + 4 x^{2} + 4 x + 2$ $C_2^8.F_5$ (as 20T317) $5120$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.42 $x^{20} + 4 x^{8} + 4 x^{5} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.43 $x^{20} + 4 x^{7} + 4 x^{5} + 4 x^{2} + 4 x + 2$ $C_2^8.F_5$ (as 20T317) $5120$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.44 $x^{20} + 4 x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.45 $x^{20} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 4 x + 2$ $C_2^8.F_5$ (as 20T317) $5120$ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.46 $x^{20} + 4 x^{8} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.47 $x^{20} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.48 $x^{20} + 4 x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{5} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.49 $x^{20} + 4 x^{3} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
2.1.20.40a1.50 $x^{20} + 4 x^{8} + 4 x^{3} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[21, 20, 0]$ $[4, 2]$ $z^{16} + z^{12} + 1,z^3 + 1$ $[5, 15, 35]$
Next   displayed columns for results