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Properties

Label 3.2.1.0a1.1-1.9.15a
Base 3.2.1.0a1.1
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: $\Q_{3}(\sqrt{2})$
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $15$
Absolute 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: $39$ (complete)
Ambiguity: $1$
Mass: $72$
Absolute Mass: $36$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.

Indices of inseparability: $[7,6,0]$ (show 34), $[7,7,0]$ (show 5)
Associated inertia: $[1]$
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 39

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Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
3.2.9.30a1.1 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{7} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[7, 7, 0]$ $[1]$ $z + (t + 1)$ undefined
3.2.9.30a1.2 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (t + 1)$ undefined
3.2.9.30a1.3 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{7} + 3 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (t + 1)$ undefined
3.2.9.30a1.4 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (t + 1)$ undefined
3.2.9.30a1.5 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{7} + 6 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (t + 1)$ undefined
3.2.9.30a1.6 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (t + 1)$ undefined
3.2.9.30a2.1 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{7} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[7, 7, 0]$ $[1]$ $z + (2 t + 2)$ undefined
3.2.9.30a2.2 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (2 t + 2)$ undefined
3.2.9.30a2.3 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{7} + 3 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (2 t + 2)$ undefined
3.2.9.30a2.4 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (2 t + 2)$ undefined
3.2.9.30a2.5 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{7} + 6 ( x^{2} + 2 x + 2 )^{6} + 3$ $F_9:C_2$ (as 18T71) $144$ $2$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (2 t + 2)$ undefined
3.2.9.30a2.6 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (2 t + 2)$ undefined
3.2.9.30a3.1 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{7} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 7, 0]$ $[1]$ $z + (2 t + 1)$ undefined
3.2.9.30a3.2 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (2 t + 1)$ undefined
3.2.9.30a3.3 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{7} + 6 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (2 t + 1)$ undefined
3.2.9.30a3.4 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{7} + 3 ( x^{2} + 2 x + 2 )^{6} + 3$ $C_3^2:F_9.C_8$ (as 18T470) $5184$ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (2 t + 1)$ undefined
3.2.9.30a3.5 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (2 t + 1)$ undefined
3.2.9.30a3.6 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{7} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (2 t + 1)$ undefined
3.2.9.30a3.7 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{7} + 6 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (2 t + 1)$ undefined
3.2.9.30a3.8 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (2 t + 1)$ undefined
3.2.9.30a3.9 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{7} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (2 t + 1)$ undefined
3.2.9.30a4.1 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 7, 0]$ $[1]$ $z + 2$ undefined
3.2.9.30a4.2 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + 2$ undefined
3.2.9.30a4.3 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + 2$ undefined
3.2.9.30a4.4 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + 2$ undefined
3.2.9.30a4.5 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + 2$ undefined
3.2.9.30a4.6 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + 2$ undefined
3.2.9.30a4.7 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 6 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + 2$ undefined
3.2.9.30a4.8 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + 2$ undefined
3.2.9.30a4.9 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + 2$ undefined
3.2.9.30a5.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{7} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 7, 0]$ $[1]$ $z + (t + 2)$ undefined
3.2.9.30a5.2 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (t + 2)$ undefined
3.2.9.30a5.3 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{7} + 6 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (t + 2)$ undefined
3.2.9.30a5.4 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{7} + 3 ( x^{2} + 2 x + 2 )^{6} + 3$ $C_3^2:F_9.C_8$ (as 18T470) $5184$ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (t + 2)$ undefined
3.2.9.30a5.5 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (t + 2)$ undefined
3.2.9.30a5.6 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{7} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (t + 2)$ undefined
3.2.9.30a5.7 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{7} + 6 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (t + 2)$ undefined
3.2.9.30a5.8 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (t + 2)$ undefined
3.2.9.30a5.9 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{7} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[7, 6, 0]$ $[1]$ $z + (t + 2)$ undefined
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