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Properties

Label 2.1.20.30b
Base 2.1.1.0a1.1
Degree \(20\)
e \(20\)
f \(1\)
c \(30\)

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Defining polynomial

$x^{20} + 2 b_{19} x^{19} + 2 b_{17} x^{17} + 2 b_{15} x^{15} + 2 b_{13} x^{13} + 2 a_{11} x^{11} + 2 c_{4} x^{4} + 2 a_{2} x^{2} + 4 c_{20} + 2$

Invariants

Residue field characteristic: $2$
Degree: $20$
Base field: $\Q_{2}$
Ramification index $e$: $20$
Residue field degree $f$: $1$
Discriminant exponent $c$: $30$
Artin slopes: $[\frac{6}{5},2]$
Swan slopes: $[\frac{1}{5},1]$
Means: $\langle\frac{1}{10},\frac{11}{20}\rangle$
Rams: $(1,9)$
Field count: $40$ (complete)
Ambiguity: $4$
Mass: $16$
Absolute Mass: $16$

Diagrams

Varying

Indices of inseparability: $[11,2,0]$
Associated inertia: $[4,1,1]$
Jump Set: $[5,11,22]$ (show 2), $[5,11,33]$ (show 16), $[5,11,35]$ (show 12), $[5,11,37]$ (show 6), $[5,11,39]$ (show 2), $[5,11,40]$ (show 2)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 12

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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
2.1.20.30b1.1 $x^{20} + 2 x^{11} + 2 x^{2} + 2$ $C_2^9.F_5$ (as 20T422) $10240$ $4$ not computed not computed not computed not computed $[11, 2, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 11, 35]$
2.1.20.30b1.2 $x^{20} + 2 x^{11} + 2 x^{2} + 6$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[11, 2, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 11, 35]$
2.1.20.30b1.3 $x^{20} + 2 x^{19} + 2 x^{11} + 2 x^{2} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[11, 2, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 11, 35]$
2.1.20.30b1.4 $x^{20} + 2 x^{17} + 2 x^{11} + 2 x^{2} + 2$ $C_2\times C_4^4:F_5$ (as 20T447) $10240$ $2$ not computed not computed not computed not computed $[11, 2, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 11, 35]$
2.1.20.30b1.5 $x^{20} + 2 x^{19} + 2 x^{17} + 2 x^{11} + 2 x^{2} + 2$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[11, 2, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 11, 35]$
2.1.20.30b1.6 $x^{20} + 2 x^{19} + 2 x^{17} + 2 x^{11} + 2 x^{2} + 6$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[11, 2, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 11, 35]$
2.1.20.30b1.21 $x^{20} + 2 x^{11} + 2 x^{4} + 2 x^{2} + 2$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[11, 2, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 11, 35]$
2.1.20.30b1.22 $x^{20} + 2 x^{19} + 2 x^{11} + 2 x^{4} + 2 x^{2} + 2$ $C_2^9.F_5$ (as 20T406) $10240$ $4$ not computed not computed not computed not computed $[11, 2, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 11, 35]$
2.1.20.30b1.23 $x^{20} + 2 x^{19} + 2 x^{11} + 2 x^{4} + 2 x^{2} + 6$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[11, 2, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 11, 35]$
2.1.20.30b1.24 $x^{20} + 2 x^{17} + 2 x^{11} + 2 x^{4} + 2 x^{2} + 2$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[11, 2, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 11, 35]$
2.1.20.30b1.25 $x^{20} + 2 x^{17} + 2 x^{11} + 2 x^{4} + 2 x^{2} + 6$ not computed $ $not computed$ $ $4$ not computed not computed not computed not computed $[11, 2, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 11, 35]$
2.1.20.30b1.26 $x^{20} + 2 x^{19} + 2 x^{17} + 2 x^{11} + 2 x^{4} + 2 x^{2} + 2$ $C_2\times C_4^4:F_5$ (as 20T448) $10240$ $2$ not computed not computed not computed not computed $[11, 2, 0]$ $[4, 1, 1]$ $z^{16} + z^{12} + 1,z^2 + 1,z + 1$ $[5, 11, 35]$
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