Properties

Label 3.1.9.19c
Base 3.1.1.0a1.1
Degree \(9\)
e \(9\)
f \(1\)
c \(19\)

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Defining polynomial

$x^{9} + 3 a_{6} x^{6} + 9 b_{13} x^{4} + 9 a_{11} x^{2} + 9 c_{9} + 3$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}$
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $19$
Artin slopes: $[2,\frac{5}{2}]$
Swan slopes: $[1,\frac{3}{2}]$
Means: $\langle\frac{2}{3},\frac{11}{9}\rangle$
Rams: $(1,\frac{5}{2})$
Field count: $18$ (complete)
Ambiguity: $3$
Mass: $12$
Absolute Mass: $12$

Diagrams

Varying

Indices of inseparability: $[11,6,0]$
Associated inertia: $[1,1]$ (show 12), $[2,1]$ (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 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
3.1.9.19c1.1 3 $x^{9} + 3 x^{6} + 9 x^{2} + 3$ $S_3^2$ (as 9T8) $36$ $1$ $[2, \frac{5}{2}]_{2}^{2}$ $[1,\frac{3}{2}]_{2}^{2}$ $[\ ]^{2}_{2}$ $[\ ]^{2}_{2}$ $[11, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.19c1.2 3 $x^{9} + 3 x^{6} + 9 x^{4} + 9 x^{2} + 3$ $S_3^2$ (as 9T8) $36$ $1$ $[2, \frac{5}{2}]_{2}^{2}$ $[1,\frac{3}{2}]_{2}^{2}$ $[\ ]^{2}_{2}$ $[\ ]^{2}_{2}$ $[11, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.19c1.3 3 $x^{9} + 3 x^{6} + 18 x^{4} + 9 x^{2} + 3$ $S_3^2$ (as 9T8) $36$ $1$ $[2, \frac{5}{2}]_{2}^{2}$ $[1,\frac{3}{2}]_{2}^{2}$ $[\ ]^{2}_{2}$ $[\ ]^{2}_{2}$ $[11, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.19c1.4 3 $x^{9} + 3 x^{6} + 18 x^{2} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$ $[\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[11, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.19c1.5 3 $x^{9} + 3 x^{6} + 9 x^{4} + 18 x^{2} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$ $[\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[11, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.19c1.6 3 $x^{9} + 3 x^{6} + 18 x^{4} + 18 x^{2} + 3$ $C_3^2 : D_{6} $ (as 9T18) $108$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}]_{2}^{2}$ $[\frac{1}{2},1,\frac{3}{2}]_{2}^{2}$ $[\frac{3}{2}]^{2}_{2}$ $[\frac{1}{2}]^{2}_{2}$ $[11, 6, 0]$ $[2, 1]$ $z^6 + 1,z + 1$ undefined
3.1.9.19c2.1 3 $x^{9} + 6 x^{6} + 9 x^{4} + 9 x^{2} + 3$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}]_{2}$ $[\frac{1}{2},1,\frac{3}{2}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[11, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.19c2.2 9 $x^{9} + 6 x^{6} + 18 x^{2} + 3$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[2, \frac{5}{2}]_{2}$ $[1,\frac{3}{2}]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[11, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.19c2.3 9 $x^{9} + 6 x^{6} + 9 x^{4} + 18 x^{2} + 3$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[2, \frac{5}{2}]_{2}$ $[1,\frac{3}{2}]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[11, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.19c2.4 9 $x^{9} + 6 x^{6} + 18 x^{4} + 18 x^{2} + 3$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[2, \frac{5}{2}]_{2}$ $[1,\frac{3}{2}]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[11, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.19c2.5 3 $x^{9} + 6 x^{6} + 9 x^{4} + 9 x^{2} + 12$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}]_{2}$ $[\frac{1}{2},1,\frac{3}{2}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[11, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.19c2.6 9 $x^{9} + 6 x^{6} + 18 x^{2} + 12$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[2, \frac{5}{2}]_{2}$ $[1,\frac{3}{2}]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[11, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.19c2.7 9 $x^{9} + 6 x^{6} + 9 x^{4} + 18 x^{2} + 12$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[2, \frac{5}{2}]_{2}$ $[1,\frac{3}{2}]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[11, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.19c2.8 9 $x^{9} + 6 x^{6} + 18 x^{4} + 18 x^{2} + 12$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[2, \frac{5}{2}]_{2}$ $[1,\frac{3}{2}]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[11, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.19c2.9 3 $x^{9} + 6 x^{6} + 9 x^{4} + 9 x^{2} + 21$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[\frac{3}{2}, 2, \frac{5}{2}]_{2}$ $[\frac{1}{2},1,\frac{3}{2}]_{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[11, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.19c2.10 9 $x^{9} + 6 x^{6} + 18 x^{2} + 21$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[2, \frac{5}{2}]_{2}$ $[1,\frac{3}{2}]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[11, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.19c2.11 9 $x^{9} + 6 x^{6} + 9 x^{4} + 18 x^{2} + 21$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[2, \frac{5}{2}]_{2}$ $[1,\frac{3}{2}]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[11, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
3.1.9.19c2.12 9 $x^{9} + 6 x^{6} + 18 x^{4} + 18 x^{2} + 21$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[2, \frac{5}{2}]_{2}$ $[1,\frac{3}{2}]_{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[11, 6, 0]$ $[1, 1]$ $z^6 + 2,2 z + 1$ undefined
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