↑

Properties

Label 2.1.20.38d
Base 2.1.1.0a1.1
Degree \(20\)
e \(20\)
f \(1\)
c \(38\)

Related objects

Downloads

Learn more

Defining polynomial

$x^{20} + 2 a_{19} x^{19} + 2 b_{18} x^{18} + 2 b_{14} x^{14} + 2 a_{10} x^{10} + 4 c_{28} x^{8} + 4 b_{27} x^{7} + 4 b_{25} x^{5} + 4 b_{23} x^{3} + 4 b_{21} x + 4 c_{20} + 2$

Invariants

Residue field characteristic: $2$
Degree: $20$
Base field: $\Q_{2}$
Ramification index $e$: $20$
Residue field degree $f$: $1$
Discriminant exponent $c$: $38$
Artin slopes: $[2,\frac{12}{5}]$
Swan slopes: $[1,\frac{7}{5}]$
Means: $\langle\frac{1}{2},\frac{19}{20}\rangle$
Rams: $(5,9)$
Field count: $160$ (complete)
Ambiguity: $4$
Mass: $64$
Absolute Mass: $64$

Diagrams

Varying

Indices of inseparability: $[19,10,0]$
Associated inertia: $[4,1,1]$
Jump Set: $[5,10,39]$ (show 40), $[5,17,37]$ (show 80), $[5,19,38]$ (show 20), $[5,19,40]$ (show 20)

Fields

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

Showing 1-50 of 80

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
2.1.20.38d1.81 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 2$ $C_2^9.F_5$ (as 20T406) $10240$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.82 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 2$ $C_2^9.F_5$ (as 20T422) $10240$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.83 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{7} + 2$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.84 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{7} + 2$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.85 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{5} + 2$ $C_2^9.F_5$ (as 20T406) $10240$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.86 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{5} + 2$ $C_2^9.F_5$ (as 20T422) $10240$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.87 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{7} + 4 x^{5} + 2$ $C_2^9.F_5$ (as 20T406) $10240$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.88 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{7} + 4 x^{5} + 2$ $C_2^9.F_5$ (as 20T422) $10240$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.89 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{3} + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.90 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{7} + 4 x^{3} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.91 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{5} + 4 x^{3} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.92 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.93 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.94 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.95 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{5} + 4 x + 2$ $C_2^9.C_2\wr F_5$ (as 20T911) $327680$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.96 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{5} + 4 x + 2$ $C_2^9.C_2\wr F_5$ (as 20T911) $327680$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.97 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{3} + 4 x + 2$ $C_2^9.C_2\wr F_5$ (as 20T907) $327680$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.98 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.99 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{5} + 4 x^{3} + 4 x + 2$ $C_2^9.C_2\wr F_5$ (as 20T907) $327680$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.100 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{5} + 4 x^{3} + 4 x + 2$ $C_2^9.C_2\wr F_5$ (as 20T907) $327680$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.101 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 6$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.102 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 6$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.103 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{7} + 6$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.104 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{7} + 6$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.105 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{5} + 6$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.106 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{5} + 6$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.107 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{7} + 4 x^{5} + 6$ $C_2^9.F_5$ (as 20T406) $10240$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.108 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{7} + 4 x^{5} + 6$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.109 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{3} + 6$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.110 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{7} + 4 x^{3} + 6$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.111 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{5} + 4 x^{3} + 6$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.112 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 6$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.113 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x + 6$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.114 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x + 6$ $C_2^9.C_2\wr F_5$ (as 20T911) $327680$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.115 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{5} + 4 x + 6$ $C_2^9.C_2\wr F_5$ (as 20T911) $327680$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.116 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{5} + 4 x + 6$ $C_2^9.C_2\wr F_5$ (as 20T911) $327680$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.117 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{3} + 4 x + 6$ $C_2^9.C_2\wr F_5$ (as 20T907) $327680$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.118 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{3} + 4 x + 6$ $C_2^9.C_2\wr F_5$ (as 20T907) $327680$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.119 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{5} + 4 x^{3} + 4 x + 6$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.120 $x^{20} + 2 x^{19} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{5} + 4 x^{3} + 4 x + 6$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.121 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 2 x^{10} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.122 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 2 x^{10} + 4 x^{7} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.123 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 2 x^{10} + 4 x^{5} + 2$ $C_2^{10}.F_5$ (as 20T526) $20480$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.124 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 2 x^{10} + 4 x^{7} + 4 x^{5} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.125 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 2 x^{10} + 4 x^{3} + 2$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.126 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{3} + 2$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.127 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 2 x^{10} + 4 x^{7} + 4 x^{3} + 2$ $C_2^9.F_5$ (as 20T406) $10240$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.128 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{7} + 4 x^{3} + 2$ $C_2^9.F_5$ (as 20T422) $10240$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.129 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 2 x^{10} + 4 x^{5} + 4 x^{3} + 2$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
2.1.20.38d1.130 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{14} + 2 x^{10} + 4 x^{8} + 4 x^{5} + 4 x^{3} + 2$ $C_2^9.F_5$ (as 20T422) $10240$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 17, 37]$
Next   displayed columns for results