Properties

Label 3.1.9.15b
Base 3.1.1.0a1.1
Degree \(9\)
e \(9\)
f \(1\)
c \(15\)

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Defining polynomial

$x^{9} + 3 b_{8} x^{8} + 3 a_{7} x^{7} + 3 a_{3} x^{3} + 9 c_{9} + 3$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}$
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $15$
Artin slopes: $[\frac{3}{2},2]$
Swan slopes: $[\frac{1}{2},1]$
Means: $\langle\frac{1}{3},\frac{7}{9}\rangle$
Rams: $(\frac{1}{2},2)$
Field count: $24$ (complete)
Ambiguity: $3$
Mass: $12$
Absolute Mass: $12$

Diagrams

Varying

Indices of inseparability: $[7,3,0]$
Associated inertia: $[1,1]$ (show 18), $[1,2]$ (show 6)
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
3.1.9.15b1.1 4 $x^{9} + 3 x^{7} + 3 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{2}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[7, 3, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.15b1.2 4 $x^{9} + 3 x^{8} + 3 x^{7} + 3 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{2}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[7, 3, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.15b1.3 2 $x^{9} + 6 x^{8} + 3 x^{7} + 3 x^{3} + 3$ $S_3^2$ (as 9T8) $36$ $1$ $[\frac{3}{2}, 2]_{2}^{2}$ $[\frac{1}{2},1]_{2}^{2}$ $[\ ]^{2}_{2}$ $[\ ]^{2}_{2}$ $[7, 3, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.15b1.4 4 $x^{9} + 6 x^{7} + 6 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{2}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[7, 3, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.15b1.5 4 $x^{9} + 3 x^{8} + 6 x^{7} + 6 x^{3} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{2}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[7, 3, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.15b1.6 2 $x^{9} + 6 x^{8} + 6 x^{7} + 6 x^{3} + 3$ $S_3^2$ (as 9T8) $36$ $1$ $[\frac{3}{2}, 2]_{2}^{2}$ $[\frac{1}{2},1]_{2}^{2}$ $[\ ]^{2}_{2}$ $[\ ]^{2}_{2}$ $[7, 3, 0]$ $[1, 2]$ $z^3 + 1,z^2 + 1$ undefined
3.1.9.15b2.1 12 $x^{9} + 6 x^{7} + 3 x^{3} + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.2 12 $x^{9} + 6 x^{7} + 3 x^{3} + 12$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.3 12 $x^{9} + 6 x^{7} + 3 x^{3} + 21$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.4 12 $x^{9} + 3 x^{8} + 6 x^{7} + 3 x^{3} + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.5 12 $x^{9} + 3 x^{8} + 6 x^{7} + 3 x^{3} + 12$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.6 12 $x^{9} + 3 x^{8} + 6 x^{7} + 3 x^{3} + 21$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.7 6 $x^{9} + 6 x^{8} + 6 x^{7} + 3 x^{3} + 3$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[\frac{3}{2}, 2]_{2}$ $[\frac{1}{2},1]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.8 6 $x^{9} + 6 x^{8} + 6 x^{7} + 3 x^{3} + 12$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[\frac{3}{2}, 2]_{2}$ $[\frac{1}{2},1]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.9 6 $x^{9} + 6 x^{8} + 6 x^{7} + 3 x^{3} + 21$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[\frac{3}{2}, 2]_{2}$ $[\frac{1}{2},1]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.10 12 $x^{9} + 3 x^{7} + 6 x^{3} + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.11 12 $x^{9} + 3 x^{7} + 6 x^{3} + 12$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.12 12 $x^{9} + 3 x^{7} + 6 x^{3} + 21$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.13 12 $x^{9} + 3 x^{8} + 3 x^{7} + 6 x^{3} + 3$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.14 12 $x^{9} + 3 x^{8} + 3 x^{7} + 6 x^{3} + 12$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.15 12 $x^{9} + 3 x^{8} + 3 x^{7} + 6 x^{3} + 21$ $C_3 \wr S_3 $ (as 9T20) $162$ $3$ $[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},1]_{2}^{3}$ $[\frac{3}{2}]^{3}_{2}$ $[\frac{1}{2}]^{3}_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.16 6 $x^{9} + 6 x^{8} + 3 x^{7} + 6 x^{3} + 3$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[\frac{3}{2}, 2]_{2}$ $[\frac{1}{2},1]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.17 6 $x^{9} + 6 x^{8} + 3 x^{7} + 6 x^{3} + 12$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[\frac{3}{2}, 2]_{2}$ $[\frac{1}{2},1]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
3.1.9.15b2.18 6 $x^{9} + 6 x^{8} + 3 x^{7} + 6 x^{3} + 21$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[\frac{3}{2}, 2]_{2}$ $[\frac{1}{2},1]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[7, 3, 0]$ $[1, 1]$ $z^3 + 1,z^2 + 2$ undefined
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