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Properties

Label 3.1.18.37d
Base 3.1.1.0a1.1
Degree \(18\)
e \(18\)
f \(1\)
c \(37\)

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Defining polynomial

$x^{18} + 3 b_{15} x^{15} + 3 a_{12} x^{12} + 9 c_{24} x^{6} + 9 b_{23} x^{5} + 9 b_{22} x^{4} + 9 a_{20} x^{2} + 3 d_{0} + 9 c_{18}$

Invariants

Residue field characteristic: $3$
Degree: $18$
Base field: $\Q_{3}$
Ramification index $e$: $18$
Residue field degree $f$: $1$
Discriminant exponent $c$: $37$
Artin slopes: $[2,\frac{7}{3}]$
Swan slopes: $[1,\frac{4}{3}]$
Means: $\langle\frac{2}{3},\frac{10}{9}\rangle$
Rams: $(2,4)$
Field count: $240$ (complete)
Ambiguity: $18$
Mass: $108$
Absolute Mass: $108$

Diagrams

Varying

Indices of inseparability: $[20,12,0]$
Associated inertia: $[1,1,1]$ (show 90), $[1,1,2]$ (show 30), $[1,2,1]$ (show 90), $[1,2,2]$ (show 30)
Jump Set: undefined (show 120), $[1,3,27]$ (show 16), $[1,7,25]$ (show 60), $[1,8,26]$ (show 36), $[1,11,29]$ (show 8)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing 1-50 of 240

Next   displayed columns for results
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
3.1.18.37d1.1 $x^{18} + 6 x^{12} + 18 x^{2} + 3$ $C_3^2:D_6$ (as 18T42) $108$ $2$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 11, 29]$
3.1.18.37d1.2 $x^{18} + 6 x^{12} + 9 x^{5} + 18 x^{2} + 3$ $C_3^2:S_3^2$ (as 18T124) $324$ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 11, 29]$
3.1.18.37d1.3 $x^{18} + 6 x^{12} + 18 x^{2} + 12$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 3, 27]$
3.1.18.37d1.4 $x^{18} + 6 x^{12} + 9 x^{5} + 18 x^{2} + 12$ $C_3^2:S_3^2$ (as 18T124) $324$ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 3, 27]$
3.1.18.37d1.5 $x^{18} + 6 x^{12} + 18 x^{2} + 21$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 3, 27]$
3.1.18.37d1.6 $x^{18} + 6 x^{12} + 9 x^{5} + 18 x^{2} + 21$ $C_3^2:S_3^2$ (as 18T124) $324$ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 3, 27]$
3.1.18.37d1.7 $x^{18} + 3 x^{15} + 6 x^{12} + 18 x^{2} + 3$ $C_3^4:S_3^2$ (as 18T409) $2916$ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 8, 26]$
3.1.18.37d1.8 $x^{18} + 3 x^{15} + 6 x^{12} + 9 x^{5} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 8, 26]$
3.1.18.37d1.9 $x^{18} + 3 x^{15} + 6 x^{12} + 18 x^{2} + 12$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 8, 26]$
3.1.18.37d1.10 $x^{18} + 3 x^{15} + 6 x^{12} + 9 x^{5} + 18 x^{2} + 12$ $C_3^4:S_3^2$ (as 18T409) $2916$ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 8, 26]$
3.1.18.37d1.11 $x^{18} + 3 x^{15} + 6 x^{12} + 18 x^{2} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 8, 26]$
3.1.18.37d1.12 $x^{18} + 3 x^{15} + 6 x^{12} + 9 x^{5} + 18 x^{2} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 8, 26]$
3.1.18.37d1.13 $x^{18} + 6 x^{15} + 6 x^{12} + 9 x^{5} + 18 x^{2} + 3$ $C_3^4:S_3^2$ (as 18T409) $2916$ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 8, 26]$
3.1.18.37d1.14 $x^{18} + 6 x^{15} + 6 x^{12} + 9 x^{5} + 18 x^{2} + 12$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 8, 26]$
3.1.18.37d1.15 $x^{18} + 6 x^{15} + 6 x^{12} + 9 x^{5} + 18 x^{2} + 21$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 8, 26]$
3.1.18.37d1.16 $x^{18} + 3 x^{12} + 18 x^{2} + 6$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.17 $x^{18} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 6$ $C_3^2:S_3^2$ (as 18T124) $324$ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.18 $x^{18} + 3 x^{12} + 18 x^{2} + 15$ $C_3^2:D_6$ (as 18T42) $108$ $2$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.19 $x^{18} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 15$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.20 $x^{18} + 3 x^{12} + 18 x^{2} + 24$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.21 $x^{18} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 24$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.22 $x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{2} + 6$ $C_3^4:S_3^2$ (as 18T409) $2916$ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.23 $x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.24 $x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{2} + 15$ $C_3^4:S_3^2$ (as 18T409) $2916$ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.25 $x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 15$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.26 $x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{2} + 24$ $C_3^4:S_3^2$ (as 18T409) $2916$ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.27 $x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 24$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.28 $x^{18} + 6 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.29 $x^{18} + 6 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 15$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.30 $x^{18} + 6 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 24$ $C_3^4:S_3^2$ (as 18T409) $2916$ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d2.1 $x^{18} + 6 x^{12} + 9 x^{2} + 3$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 11, 29]$
3.1.18.37d2.2 $x^{18} + 6 x^{12} + 9 x^{6} + 9 x^{2} + 3$ $C_3^3:C_6$ (as 18T75) $162$ $6$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 11, 29]$
3.1.18.37d2.3 $x^{18} + 6 x^{12} + 18 x^{6} + 9 x^{2} + 3$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 11, 29]$
3.1.18.37d2.4 $x^{18} + 6 x^{12} + 9 x^{5} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 11, 29]$
3.1.18.37d2.5 $x^{18} + 6 x^{12} + 9 x^{6} + 9 x^{5} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 11, 29]$
3.1.18.37d2.6 $x^{18} + 6 x^{12} + 18 x^{6} + 9 x^{5} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 11, 29]$
3.1.18.37d2.7 $x^{18} + 6 x^{12} + 9 x^{2} + 12$ $C_3^3:C_6$ (as 18T75) $162$ $6$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 3, 27]$
3.1.18.37d2.8 $x^{18} + 6 x^{12} + 9 x^{6} + 9 x^{2} + 12$ $C_3^3:C_6$ (as 18T75) $162$ $6$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 3, 27]$
3.1.18.37d2.9 $x^{18} + 6 x^{12} + 18 x^{6} + 9 x^{2} + 12$ $C_3^3:C_6$ (as 18T75) $162$ $6$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 3, 27]$
3.1.18.37d2.10 $x^{18} + 6 x^{12} + 9 x^{5} + 9 x^{2} + 12$ $C_3^4:C_6$ (as 18T163) $486$ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 3, 27]$
3.1.18.37d2.11 $x^{18} + 6 x^{12} + 9 x^{6} + 9 x^{5} + 9 x^{2} + 12$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 3, 27]$
3.1.18.37d2.12 $x^{18} + 6 x^{12} + 18 x^{6} + 9 x^{5} + 9 x^{2} + 12$ $C_3^4:C_6$ (as 18T163) $486$ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 3, 27]$
3.1.18.37d2.13 $x^{18} + 6 x^{12} + 9 x^{2} + 21$ $C_3^3:C_6$ (as 18T75) $162$ $6$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 3, 27]$
3.1.18.37d2.14 $x^{18} + 6 x^{12} + 9 x^{6} + 9 x^{2} + 21$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 3, 27]$
3.1.18.37d2.15 $x^{18} + 6 x^{12} + 18 x^{6} + 9 x^{2} + 21$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 3, 27]$
3.1.18.37d2.16 $x^{18} + 6 x^{12} + 9 x^{5} + 9 x^{2} + 21$ $C_3^4:C_6$ (as 18T163) $486$ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 3, 27]$
3.1.18.37d2.17 $x^{18} + 6 x^{12} + 9 x^{6} + 9 x^{5} + 9 x^{2} + 21$ $C_3^4:C_6$ (as 18T163) $486$ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 3, 27]$
3.1.18.37d2.18 $x^{18} + 6 x^{12} + 18 x^{6} + 9 x^{5} + 9 x^{2} + 21$ $C_3^4:C_6$ (as 18T163) $486$ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 3, 27]$
3.1.18.37d2.19 $x^{18} + 3 x^{15} + 6 x^{12} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 8, 26]$
3.1.18.37d2.20 $x^{18} + 3 x^{15} + 6 x^{12} + 9 x^{6} + 9 x^{2} + 3$ $\He_3^2:C_6$ (as 18T455) $4374$ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 8, 26]$
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