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Properties

Label 7.1.18.17a
Base 7.1.1.0a1.1
Degree \(18\)
e \(18\)
f \(1\)
c \(17\)

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Defining polynomial

$x^{18} + 7d_{0}$

Invariants

Residue field characteristic: $7$
Degree: $18$
Base field: $\Q_{7}$
Ramification index $e$: $18$
Residue field degree $f$: $1$
Discriminant exponent $c$: $17$
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: $[3]$
Jump Set: undefined (show 5), $[3]$ (show 1)

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
7.1.18.17a1.1 6 $x^{18} + 7$ $C_9:C_6$ (as 18T14) $54$ $6$ $[\ ]_{18}^{3}$ $[\ ]_{18}^{3}$ $[\ ]^{3}$ $[\ ]^{3}$ $[0]$ $[3]$ $z^{17} + 4 z^{16} + 6 z^{15} + 4 z^{14} + z^{13} + 2 z^{10} + z^9 + 5 z^8 + z^7 + 2 z^6 + z^3 + 4 z^2 + 6 z + 4$ $[3]$
7.1.18.17a1.2 6 $x^{18} + 14$ $C_9:C_6$ (as 18T14) $54$ $6$ $[\ ]_{18}^{3}$ $[\ ]_{18}^{3}$ $[\ ]^{3}$ $[\ ]^{3}$ $[0]$ $[3]$ $z^{17} + 4 z^{16} + 6 z^{15} + 4 z^{14} + z^{13} + 2 z^{10} + z^9 + 5 z^8 + z^7 + 2 z^6 + z^3 + 4 z^2 + 6 z + 4$ undefined
7.1.18.17a1.3 6 $x^{18} + 21$ $C_9:C_6$ (as 18T14) $54$ $6$ $[\ ]_{18}^{3}$ $[\ ]_{18}^{3}$ $[\ ]^{3}$ $[\ ]^{3}$ $[0]$ $[3]$ $z^{17} + 4 z^{16} + 6 z^{15} + 4 z^{14} + z^{13} + 2 z^{10} + z^9 + 5 z^8 + z^7 + 2 z^6 + z^3 + 4 z^2 + 6 z + 4$ undefined
7.1.18.17a1.4 6 $x^{18} + 28$ $C_9:C_6$ (as 18T14) $54$ $6$ $[\ ]_{18}^{3}$ $[\ ]_{18}^{3}$ $[\ ]^{3}$ $[\ ]^{3}$ $[0]$ $[3]$ $z^{17} + 4 z^{16} + 6 z^{15} + 4 z^{14} + z^{13} + 2 z^{10} + z^9 + 5 z^8 + z^7 + 2 z^6 + z^3 + 4 z^2 + 6 z + 4$ undefined
7.1.18.17a1.5 6 $x^{18} + 35$ $C_9:C_6$ (as 18T14) $54$ $6$ $[\ ]_{18}^{3}$ $[\ ]_{18}^{3}$ $[\ ]^{3}$ $[\ ]^{3}$ $[0]$ $[3]$ $z^{17} + 4 z^{16} + 6 z^{15} + 4 z^{14} + z^{13} + 2 z^{10} + z^9 + 5 z^8 + z^7 + 2 z^6 + z^3 + 4 z^2 + 6 z + 4$ undefined
7.1.18.17a1.6 6 $x^{18} + 42$ $C_9:C_6$ (as 18T14) $54$ $6$ $[\ ]_{18}^{3}$ $[\ ]_{18}^{3}$ $[\ ]^{3}$ $[\ ]^{3}$ $[0]$ $[3]$ $z^{17} + 4 z^{16} + 6 z^{15} + 4 z^{14} + z^{13} + 2 z^{10} + z^9 + 5 z^8 + z^7 + 2 z^6 + z^3 + 4 z^2 + 6 z + 4$ undefined
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