Properties

Label 3.1.9.22b
Base 3.1.1.0a1.1
Degree \(9\)
e \(9\)
f \(1\)
c \(22\)

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Defining polynomial

$x^{9} + 9 b_{16} x^{7} + 9 a_{14} x^{5} + 9 b_{12} x^{3} + 3$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}$
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $22$
Artin slopes: $[\frac{5}{2},\frac{17}{6}]$
Swan slopes: $[\frac{3}{2},\frac{11}{6}]$
Means: $\langle1,\frac{14}{9}\rangle$
Rams: $(\frac{3}{2},\frac{5}{2})$
Field count: $18$ (complete)
Ambiguity: $1$
Mass: $18$
Absolute Mass: $18$

Diagrams

Varying

Indices of inseparability: $[14,9,0]$
Associated inertia: $[1,1]$
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 18

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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
3.1.9.22b1.1 9 $x^{9} + 18 x^{5} + 3$ $C_3^2 : C_6$ (as 9T11) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.22b1.2 9 $x^{9} + 9 x^{7} + 18 x^{5} + 3$ $C_3^2 : C_6$ (as 9T11) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.22b1.3 9 $x^{9} + 18 x^{7} + 18 x^{5} + 3$ $C_3^2 : C_6$ (as 9T11) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.22b1.4 9 $x^{9} + 18 x^{5} + 9 x^{3} + 3$ $C_3^2 : C_6$ (as 9T11) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.22b1.5 9 $x^{9} + 9 x^{7} + 18 x^{5} + 9 x^{3} + 3$ $C_3^2 : C_6$ (as 9T11) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.22b1.6 9 $x^{9} + 18 x^{7} + 18 x^{5} + 9 x^{3} + 3$ $C_3^2 : C_6$ (as 9T11) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.22b1.7 9 $x^{9} + 18 x^{5} + 18 x^{3} + 3$ $C_3^2 : C_6$ (as 9T11) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.22b1.8 9 $x^{9} + 9 x^{7} + 18 x^{5} + 18 x^{3} + 3$ $C_3^2 : C_6$ (as 9T11) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.22b1.9 9 $x^{9} + 18 x^{7} + 18 x^{5} + 18 x^{3} + 3$ $C_3^2 : C_6$ (as 9T11) $54$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.22b2.1 9 $x^{9} + 9 x^{5} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[2]^{2}_{2}$ $[1]^{2}_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
3.1.9.22b2.2 9 $x^{9} + 9 x^{7} + 9 x^{5} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[2]^{2}_{2}$ $[1]^{2}_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
3.1.9.22b2.3 9 $x^{9} + 18 x^{7} + 9 x^{5} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[2]^{2}_{2}$ $[1]^{2}_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
3.1.9.22b2.4 9 $x^{9} + 9 x^{5} + 9 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[2]^{2}_{2}$ $[1]^{2}_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
3.1.9.22b2.5 9 $x^{9} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[2]^{2}_{2}$ $[1]^{2}_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
3.1.9.22b2.6 9 $x^{9} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[2]^{2}_{2}$ $[1]^{2}_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
3.1.9.22b2.7 9 $x^{9} + 9 x^{5} + 18 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[2]^{2}_{2}$ $[1]^{2}_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
3.1.9.22b2.8 9 $x^{9} + 9 x^{7} + 9 x^{5} + 18 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[2]^{2}_{2}$ $[1]^{2}_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
3.1.9.22b2.9 9 $x^{9} + 18 x^{7} + 9 x^{5} + 18 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[2, \frac{5}{2}, \frac{17}{6}]_{2}^{2}$ $[1,\frac{3}{2},\frac{11}{6}]_{2}^{2}$ $[2]^{2}_{2}$ $[1]^{2}_{2}$ $[14, 9, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
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