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Properties

Label 5.1.20.38a
Base 5.1.1.0a1.1
Degree \(20\)
e \(20\)
f \(1\)
c \(38\)

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Defining polynomial

$x^{20} + 5 a_{19} x^{19} + 25 b_{23} x^{3} + 25 b_{22} x^{2} + 25 b_{21} x + 5 d_{0}$

Invariants

Residue field characteristic: $5$
Degree: $20$
Base field: $\Q_{5}$
Ramification index $e$: $20$
Residue field degree $f$: $1$
Discriminant exponent $c$: $38$
Artin slopes: $[\frac{35}{16}]$
Swan slopes: $[\frac{19}{16}]$
Means: $\langle\frac{19}{20}\rangle$
Rams: $(\frac{19}{4})$
Field count: $500$ (complete)
Ambiguity: $4$
Mass: $500$
Absolute Mass: $500$

Diagrams

Varying

Indices of inseparability: $[19,0]$
Associated inertia: $[1,1]$
Jump Set: undefined (show 375), $[1,21]$ (show 125)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing 1-50 of 500

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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
5.1.20.38a1.1 $x^{20} + 5 x^{19} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.2 $x^{20} + 5 x^{19} + 25 x^{3} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.3 $x^{20} + 5 x^{19} + 25 x^{2} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.4 $x^{20} + 5 x^{19} + 25 x^{3} + 25 x^{2} + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.5 $x^{20} + 5 x^{19} + 50 x^{3} + 25 x^{2} + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.6 $x^{20} + 5 x^{19} + 50 x^{2} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.7 $x^{20} + 5 x^{19} + 25 x^{3} + 50 x^{2} + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.8 $x^{20} + 5 x^{19} + 50 x^{3} + 50 x^{2} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.9 $x^{20} + 5 x^{19} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.10 $x^{20} + 5 x^{19} + 25 x^{3} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.11 $x^{20} + 5 x^{19} + 50 x^{3} + 25 x + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.12 $x^{20} + 5 x^{19} + 75 x^{3} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.13 $x^{20} + 5 x^{19} + 100 x^{3} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.14 $x^{20} + 5 x^{19} + 25 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.15 $x^{20} + 5 x^{19} + 25 x^{3} + 25 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.16 $x^{20} + 5 x^{19} + 50 x^{3} + 25 x^{2} + 25 x + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.17 $x^{20} + 5 x^{19} + 75 x^{3} + 25 x^{2} + 25 x + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.18 $x^{20} + 5 x^{19} + 100 x^{3} + 25 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.19 $x^{20} + 5 x^{19} + 50 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.20 $x^{20} + 5 x^{19} + 25 x^{3} + 50 x^{2} + 25 x + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.21 $x^{20} + 5 x^{19} + 50 x^{3} + 50 x^{2} + 25 x + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.22 $x^{20} + 5 x^{19} + 75 x^{3} + 50 x^{2} + 25 x + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.23 $x^{20} + 5 x^{19} + 100 x^{3} + 50 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.24 $x^{20} + 5 x^{19} + 75 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.25 $x^{20} + 5 x^{19} + 25 x^{3} + 75 x^{2} + 25 x + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.26 $x^{20} + 5 x^{19} + 50 x^{3} + 75 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.27 $x^{20} + 5 x^{19} + 75 x^{3} + 75 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.28 $x^{20} + 5 x^{19} + 100 x^{3} + 75 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.29 $x^{20} + 5 x^{19} + 100 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.30 $x^{20} + 5 x^{19} + 25 x^{3} + 100 x^{2} + 25 x + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.31 $x^{20} + 5 x^{19} + 50 x^{3} + 100 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.32 $x^{20} + 5 x^{19} + 75 x^{3} + 100 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.33 $x^{20} + 5 x^{19} + 100 x^{3} + 100 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 1$ $[1, 21]$
5.1.20.38a1.34 $x^{20} + 10 x^{19} + 25 x^{3} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.35 $x^{20} + 10 x^{19} + 25 x^{2} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.36 $x^{20} + 10 x^{19} + 25 x^{3} + 25 x^{2} + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.37 $x^{20} + 10 x^{19} + 50 x^{3} + 25 x^{2} + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.38 $x^{20} + 10 x^{19} + 50 x^{2} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.39 $x^{20} + 10 x^{19} + 25 x^{3} + 50 x^{2} + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.40 $x^{20} + 10 x^{19} + 50 x^{3} + 50 x^{2} + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.41 $x^{20} + 10 x^{19} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.42 $x^{20} + 10 x^{19} + 25 x^{3} + 25 x + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.43 $x^{20} + 10 x^{19} + 50 x^{3} + 25 x + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.44 $x^{20} + 10 x^{19} + 75 x^{3} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.45 $x^{20} + 10 x^{19} + 100 x^{3} + 25 x + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.46 $x^{20} + 10 x^{19} + 25 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.47 $x^{20} + 10 x^{19} + 25 x^{3} + 25 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.48 $x^{20} + 10 x^{19} + 50 x^{3} + 25 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.49 $x^{20} + 10 x^{19} + 75 x^{3} + 25 x^{2} + 25 x + 5$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
5.1.20.38a1.50 $x^{20} + 10 x^{19} + 100 x^{3} + 25 x^{2} + 25 x + 5$ $C_5^4:C_{16}:C_4$ (as 20T605) $40000$ $1$ not computed not computed not computed not computed $[19, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z + 2$ $[1, 21]$
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