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Properties

Label 3.2.1.0a1.1-1.9.12a
Base 3.2.1.0a1.1
Degree \(9\)
e \(9\)
f \(1\)
c \(12\)

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Defining polynomial

$x^{9} + 3 a_{4} x^{4} + 3 b_{3} x^{3} + 3$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}(\sqrt{2})$
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Absolute Artin slopes: $[\frac{3}{2},\frac{3}{2}]$
Swan slopes: $[\frac{1}{2},\frac{1}{2}]$
Means: $\langle\frac{1}{3},\frac{4}{9}\rangle$
Rams: $(\frac{1}{2},\frac{1}{2})$
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: $[4,3,0]$ (show 34), $[4,4,0]$ (show 5)
Associated inertia: $[1]$ (show 2), $[2]$ (show 14), $[3]$ (show 13), $[4]$ (show 10)
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
3.2.9.24a1.1 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{4} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[4, 4, 0]$ $[1]$ $z^4 + 1$ undefined
3.2.9.24a2.1 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{4} + 3$ $S_3^2$ (as 18T9) $36$ $2$ not computed not computed not computed not computed $[4, 4, 0]$ $[1]$ $z^4 + 2$ undefined
3.2.9.24a3.1 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{4} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 4, 0]$ $[4]$ $z^4 + t$ undefined
3.2.9.24a4.1 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{4} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 4, 0]$ $[2]$ $z^4 + (t + 1)$ undefined
3.2.9.24a5.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{4} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 4, 0]$ $[4]$ $z^4 + 2 t$ undefined
3.2.9.24a6.1 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{4} + 6 ( x^{2} + 2 x + 2 )^{3} + 3$ $C_3^2:D_6$ (as 18T41) $108$ $2$ not computed not computed not computed not computed $[4, 3, 0]$ $[3]$ $z^4 + (t + 1) z + 1$ undefined
3.2.9.24a7.1 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{4} + 6 ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[4, 3, 0]$ $[2]$ $z^4 + (t + 1) z + 2$ undefined
3.2.9.24a8.1 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{4} + 6 ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[2]$ $z^4 + (t + 1) z + t$ undefined
3.2.9.24a9.1 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{4} + 6 ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[3]$ $z^4 + (t + 1) z + (t + 1)$ undefined
3.2.9.24a10.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{4} + 6 ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[4]$ $z^4 + (t + 1) z + 2 t$ undefined
3.2.9.24a11.1 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{4} + 3 ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[4, 3, 0]$ $[3]$ $z^4 + (2 t + 2) z + 1$ undefined
3.2.9.24a12.1 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{4} + 3 ( x^{2} + 2 x + 2 )^{3} + 3$ $F_9$ (as 18T28) $72$ $2$ not computed not computed not computed not computed $[4, 3, 0]$ $[2]$ $z^4 + (2 t + 2) z + 2$ undefined
3.2.9.24a13.1 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{4} + 3 ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[2]$ $z^4 + (2 t + 2) z + t$ undefined
3.2.9.24a14.1 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{4} + 3 ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[3]$ $z^4 + (2 t + 2) z + (t + 1)$ undefined
3.2.9.24a15.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{4} + 3 ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[4]$ $z^4 + (2 t + 2) z + 2 t$ undefined
3.2.9.24a16.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{4} + 6 x ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[2]$ $z^4 + (2 t + 1) z + 1$ undefined
3.2.9.24a17.1 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{4} + 6 x ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[3]$ $z^4 + (2 t + 1) z + 2$ undefined
3.2.9.24a18.1 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{4} + 6 x ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[4]$ $z^4 + (2 t + 1) z + t$ undefined
3.2.9.24a19.1 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{3} + 3$ $C_3^2:S_3^2$ (as 18T121) $324$ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[3]$ $z^4 + (2 t + 1) z + (t + 1)$ undefined
3.2.9.24a20.1 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{4} + 6 x ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[2]$ $z^4 + (2 t + 1) z + (t + 2)$ undefined
3.2.9.24a21.1 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{4} + 6 x ( x^{2} + 2 x + 2 )^{3} + 3$ $C_3^2:C_4\times S_3$ (as 18T95) $216$ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[2]$ $z^4 + (2 t + 1) z + 2 t$ undefined
3.2.9.24a22.1 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{4} + 6 x ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[4]$ $z^4 + (2 t + 1) z + (2 t + 1)$ undefined
3.2.9.24a23.1 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{4} + 6 x ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[3]$ $z^4 + (2 t + 1) z + (2 t + 2)$ undefined
3.2.9.24a24.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{4} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[3]$ $z^4 + 2 z + 1$ undefined
3.2.9.24a25.1 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{4} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[2]$ $z^4 + 2 z + 2$ undefined
3.2.9.24a26.1 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{4} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[2]$ $z^4 + 2 z + t$ undefined
3.2.9.24a27.1 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{3} + 3$ $C_3^4:C_{12}$ (as 18T231) $972$ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[3]$ $z^4 + 2 z + (t + 1)$ undefined
3.2.9.24a28.1 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{4} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[4]$ $z^4 + 2 z + (t + 2)$ undefined
3.2.9.24a29.1 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{4} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{3} + 3$ $C_3^4:C_{16}$ (as 18T280) $1296$ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[4]$ $z^4 + 2 z + 2 t$ undefined
3.2.9.24a30.1 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{4} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{3} + 3$ $C_3^2:C_4\times S_3$ (as 18T95) $216$ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[2]$ $z^4 + 2 z + (2 t + 1)$ undefined
3.2.9.24a31.1 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{4} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{3} + 3$ $C_3^4:C_{12}$ (as 18T231) $972$ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[3]$ $z^4 + 2 z + (2 t + 2)$ undefined
3.2.9.24a32.1 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{4} + 3 x ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[2]$ $z^4 + (t + 2) z + 1$ undefined
3.2.9.24a33.1 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{4} + 3 x ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[3]$ $z^4 + (t + 2) z + 2$ undefined
3.2.9.24a34.1 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{4} + 3 x ( x^{2} + 2 x + 2 )^{3} + 3$ $C_3^4:C_{16}$ (as 18T280) $1296$ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[4]$ $z^4 + (t + 2) z + t$ undefined
3.2.9.24a35.1 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{4} + 3 x ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[3]$ $z^4 + (t + 2) z + (t + 1)$ undefined
3.2.9.24a36.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{4} + 3 x ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[2]$ $z^4 + (t + 2) z + (t + 2)$ undefined
3.2.9.24a37.1 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{4} + 3 x ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[2]$ $z^4 + (t + 2) z + 2 t$ undefined
3.2.9.24a38.1 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{4} + 3 x ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[4]$ $z^4 + (t + 2) z + (2 t + 1)$ undefined
3.2.9.24a39.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[4, 3, 0]$ $[3]$ $z^4 + (t + 2) z + (2 t + 2)$ undefined
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