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Properties

Label 89.1.20.19a
Base 89.1.1.0a1.1
Degree \(20\)
e \(20\)
f \(1\)
c \(19\)

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Defining polynomial

$x^{20} + 89d_{0}$

Invariants

Residue field characteristic: $89$
Degree: $20$
Base field: $\Q_{89}$
Ramification index $e$: $20$
Residue field degree $f$: $1$
Discriminant exponent $c$: $19$
Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $4$ (complete)
Ambiguity: $4$
Mass: $1$
Absolute Mass: $1$

Varying

Indices of inseparability: $[0]$
Associated inertia: $[2]$
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 4

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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
89.1.20.19a1.1 4 $x^{20} + 89$ $C_4\times D_5$ (as 20T6) $40$ $4$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 12 z^{17} + 72 z^{16} + 39 z^{15} + 18 z^{14} + 45 z^{13} + z^{12} + 35 z^{11} + 17 z^{10} + 81 z^9 + 17 z^8 + 35 z^7 + z^6 + 45 z^5 + 18 z^4 + 39 z^3 + 72 z^2 + 12 z + 20$ undefined
89.1.20.19a1.2 4 $x^{20} + 267$ $C_4\times D_5$ (as 20T6) $40$ $4$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 12 z^{17} + 72 z^{16} + 39 z^{15} + 18 z^{14} + 45 z^{13} + z^{12} + 35 z^{11} + 17 z^{10} + 81 z^9 + 17 z^8 + 35 z^7 + z^6 + 45 z^5 + 18 z^4 + 39 z^3 + 72 z^2 + 12 z + 20$ undefined
89.1.20.19a1.3 4 $x^{20} + 801$ $C_4\times D_5$ (as 20T6) $40$ $4$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 12 z^{17} + 72 z^{16} + 39 z^{15} + 18 z^{14} + 45 z^{13} + z^{12} + 35 z^{11} + 17 z^{10} + 81 z^9 + 17 z^8 + 35 z^7 + z^6 + 45 z^5 + 18 z^4 + 39 z^3 + 72 z^2 + 12 z + 20$ undefined
89.1.20.19a1.4 4 $x^{20} + 2403$ $C_4\times D_5$ (as 20T6) $40$ $4$ $[\ ]_{20}^{2}$ $[\ ]_{20}^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[0]$ $[2]$ $z^{19} + 20 z^{18} + 12 z^{17} + 72 z^{16} + 39 z^{15} + 18 z^{14} + 45 z^{13} + z^{12} + 35 z^{11} + 17 z^{10} + 81 z^9 + 17 z^8 + 35 z^7 + z^6 + 45 z^5 + 18 z^4 + 39 z^3 + 72 z^2 + 12 z + 20$ undefined
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