Properties

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

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Defining polynomial

$x^{9} + 9 c_{15} x^{6} + 9 b_{14} x^{5} + 9 a_{13} x^{4} + 9 b_{12} x^{3} + 3$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}$
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $21$
Artin slopes: $[\frac{5}{2},\frac{8}{3}]$
Swan slopes: $[\frac{3}{2},\frac{5}{3}]$
Means: $\langle1,\frac{13}{9}\rangle$
Rams: $(\frac{3}{2},2)$
Field count: $36$ (complete)
Ambiguity: $3$
Mass: $18$
Absolute Mass: $18$

Diagrams

Varying

Indices of inseparability: $[13,9,0]$
Associated inertia: $[1,1]$ (show 27), $[1,2]$ (show 9)
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

  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
3.1.9.21b1.1 9 $x^{9} + 18 x^{4} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}^{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[13, 9, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.21b1.2 9 $x^{9} + 9 x^{5} + 18 x^{4} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}^{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[13, 9, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.21b1.3 9 $x^{9} + 18 x^{5} + 18 x^{4} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}^{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[13, 9, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.21b1.4 9 $x^{9} + 18 x^{4} + 9 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}^{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[13, 9, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.21b1.5 9 $x^{9} + 9 x^{5} + 18 x^{4} + 9 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}^{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[13, 9, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.21b1.6 9 $x^{9} + 18 x^{5} + 18 x^{4} + 9 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}^{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[13, 9, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.21b1.7 9 $x^{9} + 18 x^{4} + 18 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}^{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[13, 9, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.21b1.8 9 $x^{9} + 9 x^{5} + 18 x^{4} + 18 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}^{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[13, 9, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.21b1.9 9 $x^{9} + 18 x^{5} + 18 x^{4} + 18 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}^{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[13, 9, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.21b2.1 27 $x^{9} + 9 x^{4} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.2 27 $x^{9} + 9 x^{6} + 9 x^{4} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.3 27 $x^{9} + 18 x^{6} + 9 x^{4} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.4 27 $x^{9} + 9 x^{5} + 9 x^{4} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.5 27 $x^{9} + 9 x^{6} + 9 x^{5} + 9 x^{4} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.6 27 $x^{9} + 18 x^{6} + 9 x^{5} + 9 x^{4} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.7 27 $x^{9} + 18 x^{5} + 9 x^{4} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.8 27 $x^{9} + 9 x^{6} + 18 x^{5} + 9 x^{4} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.9 27 $x^{9} + 18 x^{6} + 18 x^{5} + 9 x^{4} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.10 27 $x^{9} + 9 x^{4} + 9 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.11 27 $x^{9} + 9 x^{6} + 9 x^{4} + 9 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.12 27 $x^{9} + 18 x^{6} + 9 x^{4} + 9 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.13 27 $x^{9} + 9 x^{5} + 9 x^{4} + 9 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.14 27 $x^{9} + 9 x^{6} + 9 x^{5} + 9 x^{4} + 9 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.15 27 $x^{9} + 18 x^{6} + 9 x^{5} + 9 x^{4} + 9 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.16 27 $x^{9} + 18 x^{5} + 9 x^{4} + 9 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.17 27 $x^{9} + 9 x^{6} + 18 x^{5} + 9 x^{4} + 9 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.18 27 $x^{9} + 18 x^{6} + 18 x^{5} + 9 x^{4} + 9 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.19 27 $x^{9} + 9 x^{4} + 18 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.20 27 $x^{9} + 9 x^{6} + 9 x^{4} + 18 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.21 27 $x^{9} + 18 x^{6} + 9 x^{4} + 18 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.22 27 $x^{9} + 9 x^{5} + 9 x^{4} + 18 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.23 27 $x^{9} + 9 x^{6} + 9 x^{5} + 9 x^{4} + 18 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.24 27 $x^{9} + 18 x^{6} + 9 x^{5} + 9 x^{4} + 18 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.25 27 $x^{9} + 18 x^{5} + 9 x^{4} + 18 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.26 27 $x^{9} + 9 x^{6} + 18 x^{5} + 9 x^{4} + 18 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.21b2.27 27 $x^{9} + 18 x^{6} + 18 x^{5} + 9 x^{4} + 18 x^{3} + 3$ $(C_3^2:C_3):C_2$ (as 9T12) $54$ $3$ $[\frac{3}{2}, \frac{5}{2}, \frac{8}{3}]_{2}$ $[\frac{1}{2},\frac{3}{2},\frac{5}{3}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[13, 9, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
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