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Properties

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

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Defining polynomial

$x^{20} + 2 a_{19} x^{19} + 2 b_{18} x^{18} + 2 b_{14} x^{14} + 4 b_{25} x^{5} + 4 b_{23} x^{3} + 4 b_{22} x^{2} + 4 b_{21} x + 2$

Invariants

Residue field characteristic: $2$
Degree: $20$
Base field: $\Q_{2}$
Ramification index $e$: $20$
Residue field degree $f$: $1$
Discriminant exponent $c$: $38$
Artin slopes: $[\frac{34}{15},\frac{34}{15}]$
Swan slopes: $[\frac{19}{15},\frac{19}{15}]$
Means: $\langle\frac{19}{30},\frac{19}{20}\rangle$
Rams: $(\frac{19}{3},\frac{19}{3})$
Field count: $64$ (complete)
Ambiguity: $1$
Mass: $64$
Absolute Mass: $64$

Diagrams

Varying

Indices of inseparability: $[19,14,0]$ (show 32), $[19,18,0]$ (show 16), $[19,19,0]$ (show 16)
Associated inertia: $[4,1]$
Jump Set: $[5,15,35]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 16

  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
2.1.20.38a1.1 $x^{20} + 2 x^{19} + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.2 $x^{20} + 2 x^{19} + 4 x^{5} + 2$ $F_{16}:C_4$ (as 20T173) $960$ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.3 $x^{20} + 2 x^{19} + 4 x^{3} + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.4 $x^{20} + 2 x^{19} + 4 x^{5} + 4 x^{3} + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.5 $x^{20} + 2 x^{19} + 4 x^{2} + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.6 $x^{20} + 2 x^{19} + 4 x^{5} + 4 x^{2} + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.7 $x^{20} + 2 x^{19} + 4 x^{3} + 4 x^{2} + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.8 $x^{20} + 2 x^{19} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.9 $x^{20} + 2 x^{19} + 4 x + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.10 $x^{20} + 2 x^{19} + 4 x^{5} + 4 x + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.11 $x^{20} + 2 x^{19} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.12 $x^{20} + 2 x^{19} + 4 x^{5} + 4 x^{3} + 4 x + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.13 $x^{20} + 2 x^{19} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.14 $x^{20} + 2 x^{19} + 4 x^{5} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.15 $x^{20} + 2 x^{19} + 4 x^{3} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
2.1.20.38a1.16 $x^{20} + 2 x^{19} + 4 x^{5} + 4 x^{3} + 4 x^{2} + 4 x + 2$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 19, 0]$ $[4, 1]$ $z^{16} + z^{12} + 1,z + 1$ $[5, 15, 35]$
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