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Properties

Label 19.1.9.8a
Base 19.1.1.0a1.1
Degree \(9\)
e \(9\)
f \(1\)
c \(8\)

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Defining polynomial

$x^{9} + 19d_{0}$

Invariants

Residue field characteristic: $19$
Degree: $9$
Base field: $\Q_{19}$
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $9$ (complete)
Ambiguity: $9$
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 9

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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
19.1.9.8a1.1 9 $x^{9} + 19$ $C_9$ (as 9T1) $9$ $9$ $[\ ]_{9}$ $[\ ]_{9}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^8 + 9 z^7 + 17 z^6 + 8 z^5 + 12 z^4 + 12 z^3 + 8 z^2 + 17 z + 9$ undefined
19.1.9.8a1.2 9 $x^{9} + 38$ $C_9$ (as 9T1) $9$ $9$ $[\ ]_{9}$ $[\ ]_{9}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^8 + 9 z^7 + 17 z^6 + 8 z^5 + 12 z^4 + 12 z^3 + 8 z^2 + 17 z + 9$ undefined
19.1.9.8a1.3 9 $x^{9} + 76$ $C_9$ (as 9T1) $9$ $9$ $[\ ]_{9}$ $[\ ]_{9}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^8 + 9 z^7 + 17 z^6 + 8 z^5 + 12 z^4 + 12 z^3 + 8 z^2 + 17 z + 9$ undefined
19.1.9.8a1.4 9 $x^{9} + 133$ $C_9$ (as 9T1) $9$ $9$ $[\ ]_{9}$ $[\ ]_{9}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^8 + 9 z^7 + 17 z^6 + 8 z^5 + 12 z^4 + 12 z^3 + 8 z^2 + 17 z + 9$ undefined
19.1.9.8a1.5 9 $x^{9} + 152$ $C_9$ (as 9T1) $9$ $9$ $[\ ]_{9}$ $[\ ]_{9}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^8 + 9 z^7 + 17 z^6 + 8 z^5 + 12 z^4 + 12 z^3 + 8 z^2 + 17 z + 9$ undefined
19.1.9.8a1.6 9 $x^{9} + 171$ $C_9$ (as 9T1) $9$ $9$ $[\ ]_{9}$ $[\ ]_{9}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^8 + 9 z^7 + 17 z^6 + 8 z^5 + 12 z^4 + 12 z^3 + 8 z^2 + 17 z + 9$ undefined
19.1.9.8a1.7 9 $x^{9} + 247$ $C_9$ (as 9T1) $9$ $9$ $[\ ]_{9}$ $[\ ]_{9}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^8 + 9 z^7 + 17 z^6 + 8 z^5 + 12 z^4 + 12 z^3 + 8 z^2 + 17 z + 9$ undefined
19.1.9.8a1.8 9 $x^{9} + 266$ $C_9$ (as 9T1) $9$ $9$ $[\ ]_{9}$ $[\ ]_{9}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^8 + 9 z^7 + 17 z^6 + 8 z^5 + 12 z^4 + 12 z^3 + 8 z^2 + 17 z + 9$ undefined
19.1.9.8a1.9 9 $x^{9} + 304$ $C_9$ (as 9T1) $9$ $9$ $[\ ]_{9}$ $[\ ]_{9}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^8 + 9 z^7 + 17 z^6 + 8 z^5 + 12 z^4 + 12 z^3 + 8 z^2 + 17 z + 9$ undefined
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