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Properties

Label 19.1.6.5a
Base 19.1.1.0a1.1
Degree \(6\)
e \(6\)
f \(1\)
c \(5\)

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Defining polynomial

$x^{6} + 19d_{0}$

Invariants

Residue field characteristic: $19$
Degree: $6$
Base field: $\Q_{19}$
Ramification index $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $5$
Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $6$ (complete)
Ambiguity: $6$
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 6

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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
19.1.6.5a1.1 6 $x^{6} + 19$ $C_6$ (as 6T1) $6$ $6$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^5 + 6 z^4 + 15 z^3 + z^2 + 15 z + 6$ undefined
19.1.6.5a1.2 6 $x^{6} + 38$ $C_6$ (as 6T1) $6$ $6$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^5 + 6 z^4 + 15 z^3 + z^2 + 15 z + 6$ undefined
19.1.6.5a1.3 6 $x^{6} + 76$ $C_6$ (as 6T1) $6$ $6$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^5 + 6 z^4 + 15 z^3 + z^2 + 15 z + 6$ undefined
19.1.6.5a1.4 6 $x^{6} + 152$ $C_6$ (as 6T1) $6$ $6$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^5 + 6 z^4 + 15 z^3 + z^2 + 15 z + 6$ undefined
19.1.6.5a1.5 6 $x^{6} + 247$ $C_6$ (as 6T1) $6$ $6$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^5 + 6 z^4 + 15 z^3 + z^2 + 15 z + 6$ undefined
19.1.6.5a1.6 6 $x^{6} + 304$ $C_6$ (as 6T1) $6$ $6$ $[\ ]_{6}$ $[\ ]_{6}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^5 + 6 z^4 + 15 z^3 + z^2 + 15 z + 6$ undefined
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