Properties

Label 61.1.20.19a
Base 61.1.1.0a1.1
Degree \(20\)
e \(20\)
f \(1\)
c \(19\)

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Defining polynomial

$x^{20} + 61d_{0}$

Invariants

Residue field characteristic: $61$
Degree: $20$
Base field: $\Q_{61}$
Ramification index $e$: $20$
Residue field degree $f$: $1$
Discriminant exponent $c$: $19$
Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $20$ (complete)
Ambiguity: $20$
Mass: $1$
Absolute Mass: $1$

Varying

Indices of inseparability: $[0]$
Associated inertia: $[1]$
Jump Set: undefined

Galois groups and Hidden Artin slopes

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing all 20

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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
61.1.20.19a1.1 20 $x^{20} + 61$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.2 20 $x^{20} + 122$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.3 20 $x^{20} + 183$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.4 20 $x^{20} + 244$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.5 20 $x^{20} + 366$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.6 20 $x^{20} + 488$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.7 20 $x^{20} + 549$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.8 20 $x^{20} + 671$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.9 20 $x^{20} + 732$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.10 20 $x^{20} + 976$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.11 20 $x^{20} + 1098$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.12 20 $x^{20} + 1342$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.13 20 $x^{20} + 1464$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.14 20 $x^{20} + 1647$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.15 20 $x^{20} + 1952$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.16 20 $x^{20} + 2135$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.17 20 $x^{20} + 2196$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.18 20 $x^{20} + 2684$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.19 20 $x^{20} + 2928$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
61.1.20.19a1.20 20 $x^{20} + 3294$ $C_{20}$ (as 20T1) $20$ $20$ $[\ ]_{20}$ $[\ ]_{20}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^{19} + 20 z^{18} + 7 z^{17} + 42 z^{16} + 26 z^{15} + 10 z^{14} + 25 z^{13} + 50 z^{12} + 5 z^{11} + 27 z^{10} + 48 z^9 + 27 z^8 + 5 z^7 + 50 z^6 + 25 z^5 + 10 z^4 + 26 z^3 + 42 z^2 + 7 z + 20$ undefined
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