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Properties

Label 3.1.18.31a
Base 3.1.1.0a1.1
Degree \(18\)
e \(18\)
f \(1\)
c \(31\)

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Defining polynomial

$x^{18} + 3 b_{15} x^{15} + 3 a_{14} x^{14} + 3 b_{12} x^{12} + 3 d_{0}$

Invariants

Residue field characteristic: $3$
Degree: $18$
Base field: $\Q_{3}$
Ramification index $e$: $18$
Residue field degree $f$: $1$
Discriminant exponent $c$: $31$
Artin slopes: $[\frac{15}{8},\frac{15}{8}]$
Swan slopes: $[\frac{7}{8},\frac{7}{8}]$
Means: $\langle\frac{7}{12},\frac{7}{9}\rangle$
Rams: $(\frac{7}{4},\frac{7}{4})$
Field count: $24$ (complete)
Ambiguity: $2$
Mass: $18$
Absolute Mass: $18$

Diagrams

Varying

Indices of inseparability: $[14,12,0]$ (show 16), $[14,14,0]$ (show 8)
Associated inertia: $[1,1]$ (show 12), $[1,2]$ (show 12)
Jump Set: undefined (show 12), $[1,3,23]$ (show 4), $[1,7,23]$ (show 8)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 24

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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
3.1.18.31a1.1 $x^{18} + 3 x^{14} + 3$ $F_9:C_2$ (as 18T73) $144$ $2$ not computed not computed not computed not computed $[14, 14, 0]$ $[1, 1]$ $z^9 + 2,2 z^2 + 1$ $[1, 7, 23]$
3.1.18.31a1.2 $x^{18} + 3 x^{15} + 3 x^{14} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 14, 0]$ $[1, 1]$ $z^9 + 2,2 z^2 + 1$ $[1, 7, 23]$
3.1.18.31a1.3 $x^{18} + 3 x^{14} + 3 x^{12} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 1]$ $z^9 + 2,2 z^2 + 1$ $[1, 7, 23]$
3.1.18.31a1.4 $x^{18} + 3 x^{15} + 3 x^{14} + 3 x^{12} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 1]$ $z^9 + 2,2 z^2 + 1$ $[1, 7, 23]$
3.1.18.31a1.5 $x^{18} + 3 x^{14} + 6 x^{12} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 1]$ $z^9 + 2,2 z^2 + 1$ $[1, 3, 23]$
3.1.18.31a1.6 $x^{18} + 3 x^{15} + 3 x^{14} + 6 x^{12} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 1]$ $z^9 + 2,2 z^2 + 1$ $[1, 3, 23]$
3.1.18.31a1.7 $x^{18} + 6 x^{14} + 6$ $F_9:C_2$ (as 18T73) $144$ $2$ not computed not computed not computed not computed $[14, 14, 0]$ $[1, 1]$ $z^9 + 2,2 z^2 + 1$ undefined
3.1.18.31a1.8 $x^{18} + 3 x^{15} + 6 x^{14} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 14, 0]$ $[1, 1]$ $z^9 + 2,2 z^2 + 1$ undefined
3.1.18.31a1.9 $x^{18} + 6 x^{14} + 3 x^{12} + 6$ $F_9:C_2$ (as 18T73) $144$ $2$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 1]$ $z^9 + 2,2 z^2 + 1$ undefined
3.1.18.31a1.10 $x^{18} + 3 x^{15} + 6 x^{14} + 3 x^{12} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 1]$ $z^9 + 2,2 z^2 + 1$ undefined
3.1.18.31a1.11 $x^{18} + 6 x^{14} + 6 x^{12} + 6$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 1]$ $z^9 + 2,2 z^2 + 1$ undefined
3.1.18.31a1.12 $x^{18} + 3 x^{15} + 6 x^{14} + 6 x^{12} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 1]$ $z^9 + 2,2 z^2 + 1$ undefined
3.1.18.31a2.1 $x^{18} + 6 x^{14} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[14, 14, 0]$ $[1, 2]$ $z^9 + 2,2 z^2 + 2$ $[1, 7, 23]$
3.1.18.31a2.2 $x^{18} + 3 x^{15} + 6 x^{14} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 14, 0]$ $[1, 2]$ $z^9 + 2,2 z^2 + 2$ $[1, 7, 23]$
3.1.18.31a2.3 $x^{18} + 6 x^{14} + 3 x^{12} + 3$ $F_9:C_2$ (as 18T68) $144$ $2$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 2]$ $z^9 + 2,2 z^2 + 2$ $[1, 7, 23]$
3.1.18.31a2.4 $x^{18} + 3 x^{15} + 6 x^{14} + 3 x^{12} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 2]$ $z^9 + 2,2 z^2 + 2$ $[1, 7, 23]$
3.1.18.31a2.5 $x^{18} + 6 x^{14} + 6 x^{12} + 3$ $F_9:C_2$ (as 18T68) $144$ $2$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 2]$ $z^9 + 2,2 z^2 + 2$ $[1, 3, 23]$
3.1.18.31a2.6 $x^{18} + 3 x^{15} + 6 x^{14} + 6 x^{12} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 2]$ $z^9 + 2,2 z^2 + 2$ $[1, 3, 23]$
3.1.18.31a2.7 $x^{18} + 3 x^{14} + 6$ $F_9:C_2$ (as 18T68) $144$ $2$ not computed not computed not computed not computed $[14, 14, 0]$ $[1, 2]$ $z^9 + 2,2 z^2 + 2$ undefined
3.1.18.31a2.8 $x^{18} + 3 x^{15} + 3 x^{14} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 14, 0]$ $[1, 2]$ $z^9 + 2,2 z^2 + 2$ undefined
3.1.18.31a2.9 $x^{18} + 3 x^{14} + 3 x^{12} + 6$ $F_9:C_2$ (as 18T68) $144$ $2$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 2]$ $z^9 + 2,2 z^2 + 2$ undefined
3.1.18.31a2.10 $x^{18} + 3 x^{15} + 3 x^{14} + 3 x^{12} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 2]$ $z^9 + 2,2 z^2 + 2$ undefined
3.1.18.31a2.11 $x^{18} + 3 x^{14} + 6 x^{12} + 6$ $F_9:C_2$ (as 18T68) $144$ $2$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 2]$ $z^9 + 2,2 z^2 + 2$ undefined
3.1.18.31a2.12 $x^{18} + 3 x^{15} + 3 x^{14} + 6 x^{12} + 6$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[14, 12, 0]$ $[1, 2]$ $z^9 + 2,2 z^2 + 2$ undefined
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