Properties

Label 3.1.9.18c
Base 3.1.1.0a1.1
Degree \(9\)
e \(9\)
f \(1\)
c \(18\)

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Defining polynomial

$x^{9} + 3 a_{6} x^{6} + 9 c_{12} x^{3} + 9 b_{11} x^{2} + 9 a_{10} x + 9 c_{9} + 3$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}$
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $18$
Artin slopes: $[2,\frac{7}{3}]$
Swan slopes: $[1,\frac{4}{3}]$
Means: $\langle\frac{2}{3},\frac{10}{9}\rangle$
Rams: $(1,2)$
Field count: $24$ (complete)
Ambiguity: $9$
Mass: $12$
Absolute Mass: $12$

Diagrams

Varying

Indices of inseparability: $[10,6,0]$
Associated inertia: $[1,1]$ (show 9), $[1,2]$ (show 3), $[2,1]$ (show 9), $[2,2]$ (show 3)
Jump Set: undefined

Galois groups and Hidden Artin slopes

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Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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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
3.1.9.18c1.1 3 $x^{9} + 3 x^{6} + 9 x + 3$ $C_3^2 : C_6$ (as 9T11) $54$ $1$ $[2, 2, \frac{7}{3}]^{2}$ $[1,1,\frac{4}{3}]^{2}$ $[2]^{2}$ $[1]^{2}$ $[10, 6, 0]$ $[2, 2]$ $z^6 + 1,z^2 + 1$ undefined
3.1.9.18c1.2 3 $x^{9} + 3 x^{6} + 9 x^{2} + 9 x + 3$ $C_3^2 : C_6$ (as 9T11) $54$ $1$ $[2, 2, \frac{7}{3}]^{2}$ $[1,1,\frac{4}{3}]^{2}$ $[2]^{2}$ $[1]^{2}$ $[10, 6, 0]$ $[2, 2]$ $z^6 + 1,z^2 + 1$ undefined
3.1.9.18c1.3 3 $x^{9} + 3 x^{6} + 18 x^{2} + 9 x + 3$ $C_3^2 : C_6$ (as 9T11) $54$ $1$ $[2, 2, \frac{7}{3}]^{2}$ $[1,1,\frac{4}{3}]^{2}$ $[2]^{2}$ $[1]^{2}$ $[10, 6, 0]$ $[2, 2]$ $z^6 + 1,z^2 + 1$ undefined
3.1.9.18c2.1 9 $x^{9} + 3 x^{6} + 18 x + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[2, 2, \frac{7}{3}]^{6}$ $[1,1,\frac{4}{3}]^{6}$ $[2]^{6}$ $[1]^{6}$ $[10, 6, 0]$ $[2, 1]$ $z^6 + 1,z^2 + 2$ undefined
3.1.9.18c2.2 9 $x^{9} + 3 x^{6} + 9 x^{3} + 18 x + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[2, 2, \frac{7}{3}]^{6}$ $[1,1,\frac{4}{3}]^{6}$ $[2]^{6}$ $[1]^{6}$ $[10, 6, 0]$ $[2, 1]$ $z^6 + 1,z^2 + 2$ undefined
3.1.9.18c2.3 9 $x^{9} + 3 x^{6} + 18 x^{3} + 18 x + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[2, 2, \frac{7}{3}]^{6}$ $[1,1,\frac{4}{3}]^{6}$ $[2]^{6}$ $[1]^{6}$ $[10, 6, 0]$ $[2, 1]$ $z^6 + 1,z^2 + 2$ undefined
3.1.9.18c2.4 9 $x^{9} + 3 x^{6} + 9 x^{2} + 18 x + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[2, 2, \frac{7}{3}]^{6}$ $[1,1,\frac{4}{3}]^{6}$ $[2]^{6}$ $[1]^{6}$ $[10, 6, 0]$ $[2, 1]$ $z^6 + 1,z^2 + 2$ undefined
3.1.9.18c2.5 9 $x^{9} + 3 x^{6} + 9 x^{3} + 9 x^{2} + 18 x + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[2, 2, \frac{7}{3}]^{6}$ $[1,1,\frac{4}{3}]^{6}$ $[2]^{6}$ $[1]^{6}$ $[10, 6, 0]$ $[2, 1]$ $z^6 + 1,z^2 + 2$ undefined
3.1.9.18c2.6 9 $x^{9} + 3 x^{6} + 18 x^{3} + 9 x^{2} + 18 x + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[2, 2, \frac{7}{3}]^{6}$ $[1,1,\frac{4}{3}]^{6}$ $[2]^{6}$ $[1]^{6}$ $[10, 6, 0]$ $[2, 1]$ $z^6 + 1,z^2 + 2$ undefined
3.1.9.18c2.7 9 $x^{9} + 3 x^{6} + 18 x^{2} + 18 x + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[2, 2, \frac{7}{3}]^{6}$ $[1,1,\frac{4}{3}]^{6}$ $[2]^{6}$ $[1]^{6}$ $[10, 6, 0]$ $[2, 1]$ $z^6 + 1,z^2 + 2$ undefined
3.1.9.18c2.8 9 $x^{9} + 3 x^{6} + 9 x^{3} + 18 x^{2} + 18 x + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[2, 2, \frac{7}{3}]^{6}$ $[1,1,\frac{4}{3}]^{6}$ $[2]^{6}$ $[1]^{6}$ $[10, 6, 0]$ $[2, 1]$ $z^6 + 1,z^2 + 2$ undefined
3.1.9.18c2.9 9 $x^{9} + 3 x^{6} + 18 x^{3} + 18 x^{2} + 18 x + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[2, 2, \frac{7}{3}]^{6}$ $[1,1,\frac{4}{3}]^{6}$ $[2]^{6}$ $[1]^{6}$ $[10, 6, 0]$ $[2, 1]$ $z^6 + 1,z^2 + 2$ undefined
3.1.9.18c3.1 9 $x^{9} + 6 x^{6} + 9 x^{2} + 9 x + 3$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, \frac{7}{3}]^{3}$ $[1,1,\frac{4}{3}]^{3}$ $[2]^{3}$ $[1]^{3}$ $[10, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z^2 + 1$ undefined
3.1.9.18c3.2 9 $x^{9} + 6 x^{6} + 9 x^{3} + 9 x^{2} + 9 x + 3$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, \frac{7}{3}]^{3}$ $[1,1,\frac{4}{3}]^{3}$ $[2]^{3}$ $[1]^{3}$ $[10, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z^2 + 1$ undefined
3.1.9.18c3.3 9 $x^{9} + 6 x^{6} + 18 x^{3} + 9 x^{2} + 9 x + 3$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, \frac{7}{3}]^{3}$ $[1,1,\frac{4}{3}]^{3}$ $[2]^{3}$ $[1]^{3}$ $[10, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z^2 + 1$ undefined
3.1.9.18c3.4 9 $x^{9} + 6 x^{6} + 9 x^{2} + 9 x + 12$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, \frac{7}{3}]^{3}$ $[1,1,\frac{4}{3}]^{3}$ $[2]^{3}$ $[1]^{3}$ $[10, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z^2 + 1$ undefined
3.1.9.18c3.5 9 $x^{9} + 6 x^{6} + 9 x^{3} + 9 x^{2} + 9 x + 12$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, \frac{7}{3}]^{3}$ $[1,1,\frac{4}{3}]^{3}$ $[2]^{3}$ $[1]^{3}$ $[10, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z^2 + 1$ undefined
3.1.9.18c3.6 9 $x^{9} + 6 x^{6} + 18 x^{3} + 9 x^{2} + 9 x + 12$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, \frac{7}{3}]^{3}$ $[1,1,\frac{4}{3}]^{3}$ $[2]^{3}$ $[1]^{3}$ $[10, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z^2 + 1$ undefined
3.1.9.18c3.7 9 $x^{9} + 6 x^{6} + 9 x^{2} + 9 x + 21$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, \frac{7}{3}]^{3}$ $[1,1,\frac{4}{3}]^{3}$ $[2]^{3}$ $[1]^{3}$ $[10, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z^2 + 1$ undefined
3.1.9.18c3.8 9 $x^{9} + 6 x^{6} + 9 x^{3} + 9 x^{2} + 9 x + 21$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, \frac{7}{3}]^{3}$ $[1,1,\frac{4}{3}]^{3}$ $[2]^{3}$ $[1]^{3}$ $[10, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z^2 + 1$ undefined
3.1.9.18c3.9 9 $x^{9} + 6 x^{6} + 18 x^{3} + 9 x^{2} + 9 x + 21$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, \frac{7}{3}]^{3}$ $[1,1,\frac{4}{3}]^{3}$ $[2]^{3}$ $[1]^{3}$ $[10, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z^2 + 1$ undefined
3.1.9.18c4.1 3 $x^{9} + 6 x^{6} + 9 x^{2} + 18 x + 3$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[2, 2, \frac{7}{3}]^{2}$ $[1,1,\frac{4}{3}]^{2}$ $[2]^{2}$ $[1]^{2}$ $[10, 6, 0]$ $[1, 2]$ $z^6 + 2,2 z^2 + 2$ undefined
3.1.9.18c4.2 3 $x^{9} + 6 x^{6} + 9 x^{2} + 18 x + 12$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[2, 2, \frac{7}{3}]^{2}$ $[1,1,\frac{4}{3}]^{2}$ $[2]^{2}$ $[1]^{2}$ $[10, 6, 0]$ $[1, 2]$ $z^6 + 2,2 z^2 + 2$ undefined
3.1.9.18c4.3 3 $x^{9} + 6 x^{6} + 9 x^{2} + 18 x + 21$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[2, 2, \frac{7}{3}]^{2}$ $[1,1,\frac{4}{3}]^{2}$ $[2]^{2}$ $[1]^{2}$ $[10, 6, 0]$ $[1, 2]$ $z^6 + 2,2 z^2 + 2$ undefined
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