Properties

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

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Defining polynomial

$x^{9} + 3 a_{7} x^{7} + 3 b_{6} x^{6} + 3$

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: $[\frac{15}{8},\frac{15}{8}]$
Swan slopes: $[\frac{7}{8},\frac{7}{8}]$
Means: $\langle\frac{7}{12},\frac{7}{9}\rangle$
Rams: $(\frac{7}{8},\frac{7}{8})$
Field count: $6$ (complete)
Ambiguity: $1$
Mass: $6$
Absolute Mass: $6$

Diagrams

Varying

Indices of inseparability: $[7,6,0]$ (show 4), $[7,7,0]$ (show 2)
Associated inertia: $[1]$
Jump Set: undefined

Galois groups and Hidden Artin slopes

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing all 6

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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
3.1.9.15a1.1 6 $x^{9} + 3 x^{7} + 3$ $(C_3^2:C_8):C_2$ (as 9T19) $144$ $1$ $[\frac{15}{8}, \frac{15}{8}]_{8}^{2}$ $[\frac{7}{8},\frac{7}{8}]_{8}^{2}$ $[\ ]^{2}_{8}$ $[\ ]^{2}_{8}$ $[7, 7, 0]$ $[1]$ $z + 1$ undefined
3.1.9.15a1.2 6 $x^{9} + 6 x^{7} + 3$ $(C_3^2:C_8):C_2$ (as 9T19) $144$ $1$ $[\frac{15}{8}, \frac{15}{8}]_{8}^{2}$ $[\frac{7}{8},\frac{7}{8}]_{8}^{2}$ $[\ ]^{2}_{8}$ $[\ ]^{2}_{8}$ $[7, 7, 0]$ $[1]$ $z + 1$ undefined
3.1.9.15a1.3 6 $x^{9} + 3 x^{7} + 3 x^{6} + 3$ $(C_3^2:C_8):C_2$ (as 9T19) $144$ $1$ $[\frac{15}{8}, \frac{15}{8}]_{8}^{2}$ $[\frac{7}{8},\frac{7}{8}]_{8}^{2}$ $[\ ]^{2}_{8}$ $[\ ]^{2}_{8}$ $[7, 6, 0]$ $[1]$ $z + 1$ undefined
3.1.9.15a1.4 6 $x^{9} + 6 x^{7} + 3 x^{6} + 3$ $(C_3^2:C_8):C_2$ (as 9T19) $144$ $1$ $[\frac{15}{8}, \frac{15}{8}]_{8}^{2}$ $[\frac{7}{8},\frac{7}{8}]_{8}^{2}$ $[\ ]^{2}_{8}$ $[\ ]^{2}_{8}$ $[7, 6, 0]$ $[1]$ $z + 1$ undefined
3.1.9.15a1.5 6 $x^{9} + 3 x^{7} + 6 x^{6} + 3$ $(C_3^2:C_8):C_2$ (as 9T19) $144$ $1$ $[\frac{15}{8}, \frac{15}{8}]_{8}^{2}$ $[\frac{7}{8},\frac{7}{8}]_{8}^{2}$ $[\ ]^{2}_{8}$ $[\ ]^{2}_{8}$ $[7, 6, 0]$ $[1]$ $z + 1$ undefined
3.1.9.15a1.6 6 $x^{9} + 6 x^{7} + 6 x^{6} + 3$ $(C_3^2:C_8):C_2$ (as 9T19) $144$ $1$ $[\frac{15}{8}, \frac{15}{8}]_{8}^{2}$ $[\frac{7}{8},\frac{7}{8}]_{8}^{2}$ $[\ ]^{2}_{8}$ $[\ ]^{2}_{8}$ $[7, 6, 0]$ $[1]$ $z + 1$ undefined
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