Properties

Label 3.1.9.21a
Base 3.1.1.0a1.1
Degree \(9\)
e \(9\)
f \(1\)
c \(21\)

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Defining polynomial

$x^{9} + 9 b_{16} x^{7} + 3 a_{6} x^{6} + 9 b_{14} x^{5} + 9 a_{13} x^{4} + 9 c_{9} + 3$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}$
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $21$
Artin slopes: $[2,\frac{17}{6}]$
Swan slopes: $[1,\frac{11}{6}]$
Means: $\langle\frac{2}{3},\frac{13}{9}\rangle$
Rams: $(1,\frac{7}{2})$
Field count: $36$ (complete)
Ambiguity: $3$
Mass: $36$
Absolute Mass: $36$

Diagrams

Varying

Indices of inseparability: $[13,6,0]$
Associated inertia: $[1,1]$ (show 18), $[2,1]$ (show 18)
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 36

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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
3.1.9.21a1.1 9 $x^{9} + 3 x^{6} + 9 x^{4} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{5}{2}]^{2}_{2}$ $[\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.2 9 $x^{9} + 9 x^{7} + 3 x^{6} + 9 x^{4} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{5}{2}]^{2}_{2}$ $[\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.3 9 $x^{9} + 18 x^{7} + 3 x^{6} + 9 x^{4} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{5}{2}]^{2}_{2}$ $[\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.4 9 $x^{9} + 3 x^{6} + 9 x^{5} + 9 x^{4} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{5}{2}]^{2}_{2}$ $[\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.5 9 $x^{9} + 9 x^{7} + 3 x^{6} + 9 x^{5} + 9 x^{4} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{5}{2}]^{2}_{2}$ $[\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.6 9 $x^{9} + 18 x^{7} + 3 x^{6} + 9 x^{5} + 9 x^{4} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{5}{2}]^{2}_{2}$ $[\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.7 9 $x^{9} + 3 x^{6} + 18 x^{5} + 9 x^{4} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{5}{2}]^{2}_{2}$ $[\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.8 9 $x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{5} + 9 x^{4} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{5}{2}]^{2}_{2}$ $[\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.9 9 $x^{9} + 18 x^{7} + 3 x^{6} + 18 x^{5} + 9 x^{4} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{5}{2}]^{2}_{2}$ $[\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.10 9 $x^{9} + 3 x^{6} + 18 x^{4} + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{3}{2},\frac{5}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.11 9 $x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{4} + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{3}{2},\frac{5}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.12 9 $x^{9} + 18 x^{7} + 3 x^{6} + 18 x^{4} + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{3}{2},\frac{5}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.13 9 $x^{9} + 3 x^{6} + 9 x^{5} + 18 x^{4} + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{3}{2},\frac{5}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.14 9 $x^{9} + 9 x^{7} + 3 x^{6} + 9 x^{5} + 18 x^{4} + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{3}{2},\frac{5}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.15 9 $x^{9} + 18 x^{7} + 3 x^{6} + 9 x^{5} + 18 x^{4} + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{3}{2},\frac{5}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.16 9 $x^{9} + 3 x^{6} + 18 x^{5} + 18 x^{4} + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{3}{2},\frac{5}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.17 9 $x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{5} + 18 x^{4} + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{3}{2},\frac{5}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a1.18 9 $x^{9} + 18 x^{7} + 3 x^{6} + 18 x^{5} + 18 x^{4} + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[\frac{3}{2},\frac{5}{2}]^{2}_{2}$ $[\frac{1}{2},\frac{3}{2}]^{2}_{2}$ $[13, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.21a2.1 9 $x^{9} + 6 x^{6} + 9 x^{5} + 9 x^{4} + 3$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{5}{2}]_{2}$ $[\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.2 9 $x^{9} + 9 x^{7} + 6 x^{6} + 9 x^{5} + 9 x^{4} + 3$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{5}{2}]_{2}$ $[\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.3 9 $x^{9} + 18 x^{7} + 6 x^{6} + 9 x^{5} + 9 x^{4} + 3$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{5}{2}]_{2}$ $[\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.4 9 $x^{9} + 18 x^{7} + 6 x^{6} + 9 x^{5} + 18 x^{4} + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{3}{2},\frac{5}{2}]_{2}$ $[\frac{1}{2},\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.5 9 $x^{9} + 6 x^{6} + 9 x^{5} + 18 x^{4} + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{3}{2},\frac{5}{2}]_{2}$ $[\frac{1}{2},\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.6 9 $x^{9} + 9 x^{7} + 6 x^{6} + 9 x^{5} + 18 x^{4} + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{3}{2},\frac{5}{2}]_{2}$ $[\frac{1}{2},\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.7 9 $x^{9} + 6 x^{6} + 9 x^{5} + 9 x^{4} + 12$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{5}{2}]_{2}$ $[\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.8 9 $x^{9} + 9 x^{7} + 6 x^{6} + 9 x^{5} + 9 x^{4} + 12$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{5}{2}]_{2}$ $[\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.9 9 $x^{9} + 18 x^{7} + 6 x^{6} + 9 x^{5} + 9 x^{4} + 12$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{5}{2}]_{2}$ $[\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.10 9 $x^{9} + 18 x^{7} + 6 x^{6} + 9 x^{5} + 18 x^{4} + 12$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{3}{2},\frac{5}{2}]_{2}$ $[\frac{1}{2},\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.11 9 $x^{9} + 6 x^{6} + 9 x^{5} + 18 x^{4} + 12$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{3}{2},\frac{5}{2}]_{2}$ $[\frac{1}{2},\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.12 9 $x^{9} + 9 x^{7} + 6 x^{6} + 9 x^{5} + 18 x^{4} + 12$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{3}{2},\frac{5}{2}]_{2}$ $[\frac{1}{2},\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.13 9 $x^{9} + 6 x^{6} + 9 x^{5} + 9 x^{4} + 21$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{5}{2}]_{2}$ $[\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.14 9 $x^{9} + 9 x^{7} + 6 x^{6} + 9 x^{5} + 9 x^{4} + 21$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{5}{2}]_{2}$ $[\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.15 9 $x^{9} + 18 x^{7} + 6 x^{6} + 9 x^{5} + 9 x^{4} + 21$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{5}{2}]_{2}$ $[\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.16 9 $x^{9} + 18 x^{7} + 6 x^{6} + 9 x^{5} + 18 x^{4} + 21$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{3}{2},\frac{5}{2}]_{2}$ $[\frac{1}{2},\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.17 9 $x^{9} + 6 x^{6} + 9 x^{5} + 18 x^{4} + 21$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{3}{2},\frac{5}{2}]_{2}$ $[\frac{1}{2},\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.21a2.18 9 $x^{9} + 9 x^{7} + 6 x^{6} + 9 x^{5} + 18 x^{4} + 21$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[\frac{1}{2},1,\frac{3}{2},\frac{11}{6}]_{2}$ $[\frac{3}{2},\frac{5}{2}]_{2}$ $[\frac{1}{2},\frac{3}{2}]_{2}$ $[13, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
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