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Properties

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

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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
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Showing all 20

  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
2.1.20.38d1.41 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 2$ $C_2^8.(C_2\times F_5)$ (as 20T416) $10240$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.42 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{8} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.43 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{5} + 2$ $C_2^8.(C_2\times F_5)$ (as 20T416) $10240$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.44 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{8} + 4 x^{5} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.45 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{3} + 2$ $C_2^9.(C_2\times F_5)$ (as 20T514) $20480$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.46 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{8} + 4 x^{3} + 2$ $C_2^9.(C_2\times F_5)$ (as 20T514) $20480$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.47 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{5} + 4 x^{3} + 2$ $C_2^9.(C_2\times F_5)$ (as 20T514) $20480$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.48 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{8} + 4 x^{5} + 4 x^{3} + 2$ $C_2^9.(C_2\times F_5)$ (as 20T514) $20480$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.49 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.50 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{7} + 4 x + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.51 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{5} + 4 x + 2$ $C_2^6:F_5$ (as 20T195) $1280$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.52 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{7} + 4 x^{5} + 4 x + 2$ $C_2^6:F_5$ (as 20T195) $1280$ $2$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.53 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{3} + 4 x + 2$ $C_2\wr F_5$ (as 20T137) $640$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.54 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{8} + 4 x^{3} + 4 x + 2$ $C_2\wr F_5$ (as 20T131) $640$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.55 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{7} + 4 x^{3} + 4 x + 2$ $C_2\wr F_5$ (as 20T137) $640$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.56 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{8} + 4 x^{7} + 4 x^{3} + 4 x + 2$ $C_2\wr F_5$ (as 20T131) $640$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.57 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{5} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.58 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{8} + 4 x^{5} + 4 x^{3} + 4 x + 2$ $C_2\wr F_5$ (as 20T131) $640$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.59 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
2.1.20.38d1.60 $x^{20} + 2 x^{19} + 2 x^{18} + 2 x^{10} + 4 x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 4 x + 2$ $C_2\wr F_5$ (as 20T131) $640$ $4$ not computed not computed not computed not computed $[19, 10, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 19, 38]$
  displayed columns for results