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Properties

Label 3.3.1.0a1.1-1.6.7a
Base 3.3.1.0a1.1
Degree \(6\)
e \(6\)
f \(1\)
c \(7\)

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Defining polynomial

$x^{6} + 3 c_{3} x^{3} + 3 a_{2} x^{2} + 3 d_{0}$

Invariants

Residue field characteristic: $3$
Degree: $6$
Base field: 3.3.1.0a1.1
Ramification index $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $7$
Absolute Artin slopes: $[\frac{3}{2}]$
Swan slopes: $[\frac{1}{2}]$
Means: $\langle\frac{1}{3}\rangle$
Rams: $(1)$
Field count: $30$ (complete)
Ambiguity: $6$
Mass: $26$
Absolute Mass: $26/3$

Diagrams

Varying

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

Indices of inseparability: $[2,0]$
Associated inertia: $[1,1]$ (show 20), $[1,2]$ (show 10)
Jump Set: undefined (show 15), $[1,5]$ (show 15)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 30

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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
3.3.6.21a1.1 $( x^{3} + 2 x + 1 )^{6} + 3 x ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ $C_3^3:C_{18}$ (as 18T159) $486$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ undefined
3.3.6.21a1.2 $( x^{3} + 2 x + 1 )^{6} + 6 ( x^{3} + 2 x + 1 )^{3} + 3 x ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ $C_3^3:C_6$ (as 18T85) $162$ $6$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ undefined
3.3.6.21a1.3 $( x^{3} + 2 x + 1 )^{6} + 3 ( x^{3} + 2 x + 1 )^{2} + 3$ $S_3 \times C_3$ (as 18T3) $18$ $18$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 5]$
3.3.6.21a1.4 $( x^{3} + 2 x + 1 )^{6} + 3 x^{2} ( x^{3} + 2 x + 1 )^{3} + 3 ( x^{3} + 2 x + 1 )^{2} + 3$ $C_9\times S_3$ (as 18T16) $54$ $9$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 5]$
3.3.6.21a2.1 $( x^{3} + 2 x + 1 )^{6} + 6 ( x^{3} + 2 x + 1 )^{2} + 3$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^3 + 2,2 z^2 + 2$ $[1, 5]$
3.3.6.21a3.1 $( x^{3} + 2 x + 1 )^{6} + 3 x^{2} ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^3 + 2,2 z^2 + t$ undefined
3.3.6.21a3.2 $( x^{3} + 2 x + 1 )^{6} + 3 x ( x^{3} + 2 x + 1 )^{2} + 3$ $\He_3:D_6$ (as 18T125) $324$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^3 + 2,2 z^2 + t$ $[1, 5]$
3.3.6.21a4.1 $( x^{3} + 2 x + 1 )^{6} + \left(3 x^{2} + 3 x\right) ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^3 + 2,2 z^2 + (t + 1)$ undefined
3.3.6.21a5.1 $( x^{3} + 2 x + 1 )^{6} + 6 x ( x^{3} + 2 x + 1 )^{2} + 3$ $C_3^3:C_{18}$ (as 18T159) $486$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 2 t$ $[1, 5]$
3.3.6.21a5.2 $( x^{3} + 2 x + 1 )^{6} + 6 ( x^{3} + 2 x + 1 )^{3} + 6 x ( x^{3} + 2 x + 1 )^{2} + 3$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 2 t$ $[1, 5]$
3.3.6.21a6.1 $( x^{3} + 2 x + 1 )^{6} + \left(3 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ $C_3^2:C_6$ (as 18T20) $54$ $6$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + t^2$ undefined
3.3.6.21a6.2 $( x^{3} + 2 x + 1 )^{6} + 3 x ( x^{3} + 2 x + 1 )^{3} + \left(3 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ $C_3^2:C_{18}$ (as 18T82) $162$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + t^2$ undefined
3.3.6.21a6.3 $( x^{3} + 2 x + 1 )^{6} + 3 x^{2} ( x^{3} + 2 x + 1 )^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + t^2$ $[1, 5]$
3.3.6.21a6.4 $( x^{3} + 2 x + 1 )^{6} + 6 ( x^{3} + 2 x + 1 )^{3} + 3 x^{2} ( x^{3} + 2 x + 1 )^{2} + 3$ $C_3^3:C_6$ (as 18T85) $162$ $6$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + t^2$ $[1, 5]$
3.3.6.21a7.1 $( x^{3} + 2 x + 1 )^{6} + \left(3 x^{2} + 6 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ $C_3^3:C_{18}$ (as 18T159) $486$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + (t^2 + t + 1)$ undefined
3.3.6.21a7.2 $( x^{3} + 2 x + 1 )^{6} + 6 x ( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 6 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + (t^2 + t + 1)$ undefined
3.3.6.21a8.1 $( x^{3} + 2 x + 1 )^{6} + \left(3 x^{2} + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ $C_3^2:D_6$ (as 18T42) $108$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^3 + 2,2 z^2 + (t^2 + t + 2)$ undefined
3.3.6.21a9.1 $( x^{3} + 2 x + 1 )^{6} + \left(6 x^{2} + 3 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ $S_3 \times C_3$ (as 18T3) $18$ $18$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + (t^2 + 2 t)$ undefined
3.3.6.21a9.2 $( x^{3} + 2 x + 1 )^{6} + 3 ( x^{3} + 2 x + 1 )^{3} + \left(6 x^{2} + 3 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ $C_9\times S_3$ (as 18T16) $54$ $9$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + (t^2 + 2 t)$ undefined
3.3.6.21a9.3 $( x^{3} + 2 x + 1 )^{6} + \left(3 x^{2} + 6 x\right) ( x^{3} + 2 x + 1 )^{2} + 3$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + (t^2 + 2 t)$ $[1, 5]$
3.3.6.21a9.4 $( x^{3} + 2 x + 1 )^{6} + 3 ( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 6 x\right) ( x^{3} + 2 x + 1 )^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + (t^2 + 2 t)$ $[1, 5]$
3.3.6.21a10.1 $( x^{3} + 2 x + 1 )^{6} + \left(3 x^{2} + 6 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3$ $C_3^2:D_6$ (as 18T42) $108$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^3 + 2,2 z^2 + (t^2 + 2 t + 2)$ $[1, 5]$
3.3.6.21a11.1 $( x^{3} + 2 x + 1 )^{6} + \left(3 x + 3\right) ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ $C_3^3:C_6$ (as 18T85) $162$ $6$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + (2 t^2 + 2)$ undefined
3.3.6.21a11.2 $( x^{3} + 2 x + 1 )^{6} + 3 x^{2} ( x^{3} + 2 x + 1 )^{3} + \left(3 x + 3\right) ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ $C_3^3:C_{18}$ (as 18T159) $486$ $3$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + (2 t^2 + 2)$ undefined
3.3.6.21a11.3 $( x^{3} + 2 x + 1 )^{6} + \left(6 x^{2} + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + (2 t^2 + 2)$ $[1, 5]$
3.3.6.21a11.4 $( x^{3} + 2 x + 1 )^{6} + 3 x ( x^{3} + 2 x + 1 )^{3} + \left(6 x^{2} + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3$ $C_3^2:C_6$ (as 18T20) $54$ $6$ not computed not computed not computed not computed $[2, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + (2 t^2 + 2)$ $[1, 5]$
3.3.6.21a12.1 $( x^{3} + 2 x + 1 )^{6} + \left(3 x^{2} + 6 x + 3\right) ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ $S_3 \times C_6$ (as 18T6) $36$ $6$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^3 + 2,2 z^2 + (2 t^2 + t)$ undefined
3.3.6.21a12.2 $( x^{3} + 2 x + 1 )^{6} + \left(6 x^{2} + 3 x\right) ( x^{3} + 2 x + 1 )^{2} + 3$ $\He_3:D_6$ (as 18T125) $324$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^3 + 2,2 z^2 + (2 t^2 + t)$ $[1, 5]$
3.3.6.21a13.1 $( x^{3} + 2 x + 1 )^{6} + \left(3 x^{2} + 3 x + 3\right) ( x^{3} + 2 x + 1 )^{2} + 3 ( x^{3} + 2 x + 1 ) + 3 x$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^3 + 2,2 z^2 + (2 t^2 + t + 2)$ undefined
3.3.6.21a13.2 $( x^{3} + 2 x + 1 )^{6} + \left(6 x^{2} + 3 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3$ $\He_3:D_6$ (as 18T125) $324$ $2$ not computed not computed not computed not computed $[2, 0]$ $[1, 2]$ $z^3 + 2,2 z^2 + (2 t^2 + t + 2)$ $[1, 5]$
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