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Properties

Label 37.1.20.19a
Base 37.1.1.0a1.1
Degree \(20\)
e \(20\)
f \(1\)
c \(19\)

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Defining polynomial

$x^{20} + 37d_{0}$

Invariants

Residue field characteristic: $37$
Degree: $20$
Base field: $\Q_{37}$
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: $[4]$
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
37.1.20.19a1.1 4 $x^{20} + 37$ $C_4\times F_5$ (as 20T20) $80$ $4$ $[\ ]_{20}^{4}$ $[\ ]_{20}^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[0]$ $[4]$ $z^{19} + 20 z^{18} + 5 z^{17} + 30 z^{16} + 35 z^{15} + z^{14} + 21 z^{13} + 5 z^{12} + 22 z^{11} + 17 z^{10} + 15 z^9 + 17 z^8 + 22 z^7 + 5 z^6 + 21 z^5 + z^4 + 35 z^3 + 30 z^2 + 5 z + 20$ undefined
37.1.20.19a1.2 4 $x^{20} + 74$ $C_4\times F_5$ (as 20T20) $80$ $4$ $[\ ]_{20}^{4}$ $[\ ]_{20}^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[0]$ $[4]$ $z^{19} + 20 z^{18} + 5 z^{17} + 30 z^{16} + 35 z^{15} + z^{14} + 21 z^{13} + 5 z^{12} + 22 z^{11} + 17 z^{10} + 15 z^9 + 17 z^8 + 22 z^7 + 5 z^6 + 21 z^5 + z^4 + 35 z^3 + 30 z^2 + 5 z + 20$ undefined
37.1.20.19a1.3 4 $x^{20} + 148$ $C_4\times F_5$ (as 20T20) $80$ $4$ $[\ ]_{20}^{4}$ $[\ ]_{20}^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[0]$ $[4]$ $z^{19} + 20 z^{18} + 5 z^{17} + 30 z^{16} + 35 z^{15} + z^{14} + 21 z^{13} + 5 z^{12} + 22 z^{11} + 17 z^{10} + 15 z^9 + 17 z^8 + 22 z^7 + 5 z^6 + 21 z^5 + z^4 + 35 z^3 + 30 z^2 + 5 z + 20$ undefined
37.1.20.19a1.4 4 $x^{20} + 296$ $C_4\times F_5$ (as 20T20) $80$ $4$ $[\ ]_{20}^{4}$ $[\ ]_{20}^{4}$ $[\ ]^{4}$ $[\ ]^{4}$ $[0]$ $[4]$ $z^{19} + 20 z^{18} + 5 z^{17} + 30 z^{16} + 35 z^{15} + z^{14} + 21 z^{13} + 5 z^{12} + 22 z^{11} + 17 z^{10} + 15 z^9 + 17 z^8 + 22 z^7 + 5 z^6 + 21 z^5 + z^4 + 35 z^3 + 30 z^2 + 5 z + 20$ undefined
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