Properties

Label 3.1.9.13b
Base 3.1.1.0a1.1
Degree \(9\)
e \(9\)
f \(1\)
c \(13\)

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Defining polynomial

$x^{9} + 3 c_{6} x^{6} + 3 a_{5} x^{5} + 3 a_{3} x^{3} + 3$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}$
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $13$
Artin slopes: $[\frac{3}{2},\frac{5}{3}]$
Swan slopes: $[\frac{1}{2},\frac{2}{3}]$
Means: $\langle\frac{1}{3},\frac{5}{9}\rangle$
Rams: $(\frac{1}{2},1)$
Field count: $8$ (complete)
Ambiguity: $3$
Mass: $4$
Absolute Mass: $4$

Diagrams

Varying

Indices of inseparability: $[5,3,0]$
Associated inertia: $[1,1]$ (show 6), $[1,2]$ (show 2)
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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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
3.1.9.13b1.1 2 $x^{9} + 6 x^{5} + 3 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{3}]_{2}^{2}$ $[\frac{1}{2},\frac{1}{2},\frac{2}{3}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[5, 3, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.13b1.2 2 $x^{9} + 3 x^{5} + 6 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{3}]_{2}^{2}$ $[\frac{1}{2},\frac{1}{2},\frac{2}{3}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[5, 3, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.13b2.1 6 $x^{9} + 3 x^{5} + 3 x^{3} + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{3}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{2}{3}]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[5, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.13b2.2 6 $x^{9} + 3 x^{6} + 3 x^{5} + 3 x^{3} + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{3}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{2}{3}]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[5, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.13b2.3 6 $x^{9} + 6 x^{6} + 3 x^{5} + 3 x^{3} + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{3}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{2}{3}]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[5, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.13b2.4 6 $x^{9} + 6 x^{5} + 6 x^{3} + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{3}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{2}{3}]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[5, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.13b2.5 6 $x^{9} + 3 x^{6} + 6 x^{5} + 6 x^{3} + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{3}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{2}{3}]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[5, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.13b2.6 6 $x^{9} + 6 x^{6} + 6 x^{5} + 6 x^{3} + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{3}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{2}{3}]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[5, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
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