Properties

Label 3.1.9.16b
Base 3.1.1.0a1.1
Degree \(9\)
e \(9\)
f \(1\)
c \(16\)

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Defining polynomial

$x^{9} + 3 a_{8} x^{8} + 3 a_{3} x^{3} + 9 b_{10} x + 3$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}$
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
Artin slopes: $[\frac{3}{2},\frac{13}{6}]$
Swan slopes: $[\frac{1}{2},\frac{7}{6}]$
Means: $\langle\frac{1}{3},\frac{8}{9}\rangle$
Rams: $(\frac{1}{2},\frac{5}{2})$
Field count: $12$ (complete)
Ambiguity: $1$
Mass: $12$
Absolute Mass: $12$

Diagrams

Varying

Indices of inseparability: $[8,3,0]$
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 12

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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
3.1.9.16b1.1 6 $x^{9} + 3 x^{8} + 3 x^{3} + 3$ $(C_9:C_3):C_2$ (as 9T10) $54$ $1$ $[\frac{3}{2}, 2, \frac{13}{6}]_{2}$ $[\frac{1}{2},1,\frac{7}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[8, 3, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.16b1.2 6 $x^{9} + 3 x^{8} + 3 x^{3} + 9 x + 3$ $(C_9:C_3):C_2$ (as 9T10) $54$ $1$ $[\frac{3}{2}, 2, \frac{13}{6}]_{2}$ $[\frac{1}{2},1,\frac{7}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[8, 3, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.16b1.3 6 $x^{9} + 3 x^{8} + 3 x^{3} + 18 x + 3$ $(C_9:C_3):C_2$ (as 9T10) $54$ $1$ $[\frac{3}{2}, 2, \frac{13}{6}]_{2}$ $[\frac{1}{2},1,\frac{7}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[8, 3, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.16b1.4 6 $x^{9} + 3 x^{8} + 6 x^{3} + 3$ $(C_9:C_3):C_2$ (as 9T10) $54$ $1$ $[\frac{3}{2}, 2, \frac{13}{6}]_{2}$ $[\frac{1}{2},1,\frac{7}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[8, 3, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.16b1.5 6 $x^{9} + 3 x^{8} + 6 x^{3} + 9 x + 3$ $(C_9:C_3):C_2$ (as 9T10) $54$ $1$ $[\frac{3}{2}, 2, \frac{13}{6}]_{2}$ $[\frac{1}{2},1,\frac{7}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[8, 3, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.16b1.6 6 $x^{9} + 3 x^{8} + 6 x^{3} + 18 x + 3$ $(C_9:C_3):C_2$ (as 9T10) $54$ $1$ $[\frac{3}{2}, 2, \frac{13}{6}]_{2}$ $[\frac{1}{2},1,\frac{7}{6}]_{2}$ $[2]_{2}$ $[1]_{2}$ $[8, 3, 0]$ $[1, 1]$ $z^3 + 1,z + 1$ undefined
3.1.9.16b2.1 6 $x^{9} + 6 x^{8} + 3 x^{3} + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, \frac{3}{2}, 2, \frac{13}{6}]_{2}^{2}$ $[\frac{1}{2},\frac{1}{2},1,\frac{7}{6}]_{2}^{2}$ $[\frac{3}{2},2]^{2}_{2}$ $[\frac{1}{2},1]^{2}_{2}$ $[8, 3, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
3.1.9.16b2.2 6 $x^{9} + 6 x^{8} + 3 x^{3} + 9 x + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, \frac{3}{2}, 2, \frac{13}{6}]_{2}^{2}$ $[\frac{1}{2},\frac{1}{2},1,\frac{7}{6}]_{2}^{2}$ $[\frac{3}{2},2]^{2}_{2}$ $[\frac{1}{2},1]^{2}_{2}$ $[8, 3, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
3.1.9.16b2.3 6 $x^{9} + 6 x^{8} + 3 x^{3} + 18 x + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, \frac{3}{2}, 2, \frac{13}{6}]_{2}^{2}$ $[\frac{1}{2},\frac{1}{2},1,\frac{7}{6}]_{2}^{2}$ $[\frac{3}{2},2]^{2}_{2}$ $[\frac{1}{2},1]^{2}_{2}$ $[8, 3, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
3.1.9.16b2.4 6 $x^{9} + 6 x^{8} + 6 x^{3} + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, \frac{3}{2}, 2, \frac{13}{6}]_{2}^{2}$ $[\frac{1}{2},\frac{1}{2},1,\frac{7}{6}]_{2}^{2}$ $[\frac{3}{2},2]^{2}_{2}$ $[\frac{1}{2},1]^{2}_{2}$ $[8, 3, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
3.1.9.16b2.5 6 $x^{9} + 6 x^{8} + 6 x^{3} + 9 x + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, \frac{3}{2}, 2, \frac{13}{6}]_{2}^{2}$ $[\frac{1}{2},\frac{1}{2},1,\frac{7}{6}]_{2}^{2}$ $[\frac{3}{2},2]^{2}_{2}$ $[\frac{1}{2},1]^{2}_{2}$ $[8, 3, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
3.1.9.16b2.6 6 $x^{9} + 6 x^{8} + 6 x^{3} + 18 x + 3$ $((C_3^3:C_3):C_2):C_2$ (as 9T24) $324$ $1$ $[\frac{3}{2}, \frac{3}{2}, 2, \frac{13}{6}]_{2}^{2}$ $[\frac{1}{2},\frac{1}{2},1,\frac{7}{6}]_{2}^{2}$ $[\frac{3}{2},2]^{2}_{2}$ $[\frac{1}{2},1]^{2}_{2}$ $[8, 3, 0]$ $[1, 1]$ $z^3 + 1,z + 2$ undefined
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