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Properties

Label 3.1.18.27c
Base 3.1.1.0a1.1
Degree \(18\)
e \(18\)
f \(1\)
c \(27\)

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Defining polynomial

$x^{18} + 3 c_{12} x^{12} + 3 b_{11} x^{11} + 3 a_{10} x^{10} + 3 c_{9} x^{9} + 3 a_{6} x^{6} + 3 d_{0}$

Invariants

Residue field characteristic: $3$
Degree: $18$
Base field: $\Q_{3}$
Ramification index $e$: $18$
Residue field degree $f$: $1$
Discriminant exponent $c$: $27$
Artin slopes: $[\frac{3}{2},\frac{5}{3}]$
Swan slopes: $[\frac{1}{2},\frac{2}{3}]$
Means: $\langle\frac{1}{3},\frac{5}{9}\rangle$
Rams: $(1,2)$
Field count: $32$ (complete)
Ambiguity: $18$
Mass: $12$
Absolute Mass: $12$

Diagrams

Varying

Indices of inseparability: $[10,6,0]$
Associated inertia: $[1,1,1]$ (show 12), $[1,1,2]$ (show 4), $[1,2,1]$ (show 12), $[1,2,2]$ (show 4)
Jump Set: undefined (show 16), $[1,5,19]$ (show 16)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 32

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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
3.1.18.27c1.1 $x^{18} + 6 x^{10} + 3 x^{6} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 19]$
3.1.18.27c1.2 $x^{18} + 6 x^{10} + 3 x^{9} + 3 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 19]$
3.1.18.27c1.3 $x^{18} + 3 x^{10} + 6 x^{6} + 6$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.27c1.4 $x^{18} + 3 x^{10} + 3 x^{9} + 6 x^{6} + 6$ $C_3^4:D_6$ (as 18T239) $972$ $1$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.27c2.1 $x^{18} + 3 x^{10} + 3 x^{6} + 3$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 19]$
3.1.18.27c2.2 $x^{18} + 3 x^{12} + 3 x^{10} + 3 x^{6} + 3$ $C_3\wr S_3$ (as 18T86) $162$ $6$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 19]$
3.1.18.27c2.3 $x^{18} + 6 x^{12} + 3 x^{10} + 3 x^{6} + 3$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 19]$
3.1.18.27c2.4 $x^{18} + 3 x^{10} + 3 x^{9} + 3 x^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 19]$
3.1.18.27c2.5 $x^{18} + 3 x^{12} + 3 x^{10} + 3 x^{9} + 3 x^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 19]$
3.1.18.27c2.6 $x^{18} + 6 x^{12} + 3 x^{10} + 3 x^{9} + 3 x^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 19]$
3.1.18.27c2.7 $x^{18} + 6 x^{10} + 6 x^{6} + 6$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.27c2.8 $x^{18} + 3 x^{12} + 6 x^{10} + 6 x^{6} + 6$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.27c2.9 $x^{18} + 6 x^{12} + 6 x^{10} + 6 x^{6} + 6$ $C_3\wr S_3$ (as 18T86) $162$ $6$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.27c2.10 $x^{18} + 6 x^{10} + 3 x^{9} + 6 x^{6} + 6$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.27c2.11 $x^{18} + 3 x^{12} + 6 x^{10} + 3 x^{9} + 6 x^{6} + 6$ $C_3^3.(C_3\times S_3)$ (as 18T173) $486$ $3$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.27c2.12 $x^{18} + 6 x^{12} + 6 x^{10} + 3 x^{9} + 6 x^{6} + 6$ $C_3^3.(C_3\times S_3)$ (as 18T173) $486$ $3$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.27c3.1 $x^{18} + 6 x^{10} + 6 x^{6} + 3$ $C_3^3:D_6$ (as 18T119) $324$ $6$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 1]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 1$ $[1, 5, 19]$
3.1.18.27c3.2 $x^{18} + 3 x^{12} + 6 x^{10} + 6 x^{6} + 3$ $C_3^3:D_6$ (as 18T119) $324$ $6$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 1]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 1$ $[1, 5, 19]$
3.1.18.27c3.3 $x^{18} + 6 x^{12} + 6 x^{10} + 6 x^{6} + 3$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 1]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 1$ $[1, 5, 19]$
3.1.18.27c3.4 $x^{18} + 3 x^{11} + 6 x^{10} + 6 x^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 1]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 1$ $[1, 5, 19]$
3.1.18.27c3.5 $x^{18} + 3 x^{12} + 3 x^{11} + 6 x^{10} + 6 x^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 1]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 1$ $[1, 5, 19]$
3.1.18.27c3.6 $x^{18} + 6 x^{12} + 3 x^{11} + 6 x^{10} + 6 x^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 1]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 1$ $[1, 5, 19]$
3.1.18.27c3.7 $x^{18} + 3 x^{10} + 3 x^{6} + 6$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 1]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 1$ undefined
3.1.18.27c3.8 $x^{18} + 3 x^{12} + 3 x^{10} + 3 x^{6} + 6$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 1]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 1$ undefined
3.1.18.27c3.9 $x^{18} + 6 x^{12} + 3 x^{10} + 3 x^{6} + 6$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 1]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 1$ undefined
3.1.18.27c3.10 $x^{18} + 3 x^{11} + 3 x^{10} + 3 x^{6} + 6$ $C_3^3:S_3^2$ (as 18T241) $972$ $3$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 1]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 1$ undefined
3.1.18.27c3.11 $x^{18} + 3 x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{6} + 6$ $C_3^3:S_3^2$ (as 18T241) $972$ $3$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 1]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 1$ undefined
3.1.18.27c3.12 $x^{18} + 6 x^{12} + 3 x^{11} + 3 x^{10} + 3 x^{6} + 6$ $C_3^3:S_3^2$ (as 18T241) $972$ $3$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 1]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 1$ undefined
3.1.18.27c4.1 $x^{18} + 3 x^{10} + 6 x^{6} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 2]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 2$ $[1, 5, 19]$
3.1.18.27c4.2 $x^{18} + 3 x^{11} + 3 x^{10} + 6 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 2]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 2$ $[1, 5, 19]$
3.1.18.27c4.3 $x^{18} + 6 x^{10} + 3 x^{6} + 6$ $C_3^2:D_6$ (as 18T55) $108$ $2$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 2]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 2$ undefined
3.1.18.27c4.4 $x^{18} + 3 x^{11} + 6 x^{10} + 3 x^{6} + 6$ $C_3^4:D_6$ (as 18T236) $972$ $1$ not computed not computed not computed not computed $[10, 6, 0]$ $[1, 2, 2]$ $z^9 + 2,2 z^6 + 2,2 z^2 + 2$ undefined
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