Properties

Label 5.1.20.35a
Base 5.1.1.0a1.1
Degree \(20\)
e \(20\)
f \(1\)
c \(35\)

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Defining polynomial

$x^{20} + 5 b_{19} x^{19} + 5 b_{18} x^{18} + 5 b_{17} x^{17} + 5 a_{16} x^{16} + 5 d_{0} + 25 c_{20}$

Invariants

Residue field characteristic: $5$
Degree: $20$
Base field: $\Q_{5}$
Ramification index $e$: $20$
Residue field degree $f$: $1$
Discriminant exponent $c$: $35$
Artin slopes: $[2]$
Swan slopes: $[1]$
Means: $\langle\frac{4}{5}\rangle$
Rams: $(4)$
Field count: $1056$ (complete)
Ambiguity: $20$
Mass: $500$
Absolute Mass: $500$

Diagrams

Varying

Indices of inseparability: $[16,0]$
Associated inertia: $[1,1]$ (show 660), $[1,2]$ (show 132), $[1,4]$ (show 264)
Jump Set: undefined (show 792), $[1,5]$ (show 1), $[1,21]$ (show 99), $[1,22]$ (show 125), $[1,23]$ (show 30), $[1,24]$ (show 5), $[1,25]$ (show 4)

Galois groups and Hidden Artin slopes

Select desired size of Galois group. Note that the following data has not all been computed for fields in this family, so the tables below are incomplete.

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing 1-50 of at least 1000

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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
5.1.20.35a1.1 $x^{20} + 20 x^{16} + 5$ $C_{20}$ (as 20T1) $20$ $20$ $[2]_{4}$ $[1]_{4}$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 5]$
5.1.20.35a1.2 $x^{20} + 20 x^{16} + 30$ $C_{20}$ (as 20T1) $20$ $20$ $[2]_{4}$ $[1]_{4}$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 25]$
5.1.20.35a1.3 $x^{20} + 20 x^{16} + 55$ $C_{20}$ (as 20T1) $20$ $20$ $[2]_{4}$ $[1]_{4}$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 25]$
5.1.20.35a1.4 $x^{20} + 20 x^{16} + 80$ $C_{20}$ (as 20T1) $20$ $20$ $[2]_{4}$ $[1]_{4}$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 25]$
5.1.20.35a1.5 $x^{20} + 20 x^{16} + 105$ $C_{20}$ (as 20T1) $20$ $20$ $[2]_{4}$ $[1]_{4}$ $[\ ]$ $[\ ]$ $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 25]$
5.1.20.35a1.6 $x^{20} + 5 x^{19} + 20 x^{16} + 5$ $C_5\times F_5$ (as 20T29) $100$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 24]$
5.1.20.35a1.7 $x^{20} + 5 x^{19} + 20 x^{16} + 30$ $C_5\times F_5$ (as 20T29) $100$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 24]$
5.1.20.35a1.8 $x^{20} + 5 x^{19} + 20 x^{16} + 55$ $C_5\times F_5$ (as 20T29) $100$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 24]$
5.1.20.35a1.9 $x^{20} + 5 x^{19} + 20 x^{16} + 80$ $C_5\times F_5$ (as 20T29) $100$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 24]$
5.1.20.35a1.10 $x^{20} + 5 x^{19} + 20 x^{16} + 105$ $C_5\times F_5$ (as 20T29) $100$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 24]$
5.1.20.35a1.11 $x^{20} + 5 x^{18} + 20 x^{16} + 5$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.12 $x^{20} + 5 x^{18} + 20 x^{16} + 30$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.13 $x^{20} + 5 x^{18} + 20 x^{16} + 55$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.14 $x^{20} + 5 x^{18} + 20 x^{16} + 80$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.15 $x^{20} + 5 x^{18} + 20 x^{16} + 105$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.16 $x^{20} + 5 x^{19} + 5 x^{18} + 20 x^{16} + 5$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.17 $x^{20} + 5 x^{19} + 5 x^{18} + 20 x^{16} + 30$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.18 $x^{20} + 5 x^{19} + 5 x^{18} + 20 x^{16} + 55$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.19 $x^{20} + 5 x^{19} + 5 x^{18} + 20 x^{16} + 80$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.20 $x^{20} + 5 x^{19} + 5 x^{18} + 20 x^{16} + 105$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.21 $x^{20} + 10 x^{19} + 5 x^{18} + 20 x^{16} + 5$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.22 $x^{20} + 10 x^{19} + 5 x^{18} + 20 x^{16} + 30$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.23 $x^{20} + 10 x^{19} + 5 x^{18} + 20 x^{16} + 55$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.24 $x^{20} + 10 x^{19} + 5 x^{18} + 20 x^{16} + 80$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.25 $x^{20} + 10 x^{19} + 5 x^{18} + 20 x^{16} + 105$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.26 $x^{20} + 10 x^{18} + 20 x^{16} + 5$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.27 $x^{20} + 10 x^{18} + 20 x^{16} + 30$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.28 $x^{20} + 10 x^{18} + 20 x^{16} + 55$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.29 $x^{20} + 10 x^{18} + 20 x^{16} + 80$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.30 $x^{20} + 10 x^{18} + 20 x^{16} + 105$ $C_5:C_{20}$ (as 20T25) $100$ $10$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.31 $x^{20} + 5 x^{19} + 10 x^{18} + 20 x^{16} + 5$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.32 $x^{20} + 5 x^{19} + 10 x^{18} + 20 x^{16} + 30$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.33 $x^{20} + 5 x^{19} + 10 x^{18} + 20 x^{16} + 55$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.34 $x^{20} + 5 x^{19} + 10 x^{18} + 20 x^{16} + 80$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.35 $x^{20} + 5 x^{19} + 10 x^{18} + 20 x^{16} + 105$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.36 $x^{20} + 10 x^{19} + 10 x^{18} + 20 x^{16} + 5$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.37 $x^{20} + 10 x^{19} + 10 x^{18} + 20 x^{16} + 30$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.38 $x^{20} + 10 x^{19} + 10 x^{18} + 20 x^{16} + 55$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.39 $x^{20} + 10 x^{19} + 10 x^{18} + 20 x^{16} + 80$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.40 $x^{20} + 10 x^{19} + 10 x^{18} + 20 x^{16} + 105$ $C_5^2:C_{20}$ (as 20T127) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 23]$
5.1.20.35a1.41 $x^{20} + 5 x^{17} + 20 x^{16} + 5$ $C_5\times F_5$ (as 20T29) $100$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 22]$
5.1.20.35a1.42 $x^{20} + 5 x^{17} + 20 x^{16} + 30$ $C_5\times F_5$ (as 20T29) $100$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 22]$
5.1.20.35a1.43 $x^{20} + 5 x^{17} + 20 x^{16} + 55$ $C_5\times F_5$ (as 20T29) $100$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 22]$
5.1.20.35a1.44 $x^{20} + 5 x^{17} + 20 x^{16} + 80$ $C_5\times F_5$ (as 20T29) $100$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 22]$
5.1.20.35a1.45 $x^{20} + 5 x^{17} + 20 x^{16} + 105$ $C_5\times F_5$ (as 20T29) $100$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 22]$
5.1.20.35a1.46 $x^{20} + 5 x^{19} + 5 x^{17} + 20 x^{16} + 5$ $C_5^2:C_{20}$ (as 20T126) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 22]$
5.1.20.35a1.47 $x^{20} + 5 x^{19} + 5 x^{17} + 20 x^{16} + 30$ $C_5^2:C_{20}$ (as 20T126) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 22]$
5.1.20.35a1.48 $x^{20} + 5 x^{19} + 5 x^{17} + 20 x^{16} + 55$ $C_5^2:C_{20}$ (as 20T126) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 22]$
5.1.20.35a1.49 $x^{20} + 5 x^{19} + 5 x^{17} + 20 x^{16} + 80$ $C_5^2:C_{20}$ (as 20T126) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 22]$
5.1.20.35a1.50 $x^{20} + 5 x^{19} + 5 x^{17} + 20 x^{16} + 105$ $C_5^2:C_{20}$ (as 20T126) $500$ $5$ not computed not computed not computed not computed $[16, 0]$ $[1, 1]$ $z^{15} + 4 z^{10} + z^5 + 4,4 z^4 + 1$ $[1, 22]$
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