Properties

Label 3.2.1.0a1.1-1.9.16a
Base 3.2.1.0a1.1
Degree \(9\)
e \(9\)
f \(1\)
c \(16\)

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Defining polynomial

$x^{9} + 3 a_{8} x^{8} + 3 b_{6} x^{6} + 9 c_{9} + 3$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}(\sqrt{2})$
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
Absolute Artin slopes: $[2,2]$
Swan slopes: $[1,1]$
Means: $\langle\frac{2}{3},\frac{8}{9}\rangle$
Rams: $(1,1)$
Field count: $73$ (complete)
Ambiguity: $9$
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: $[8,6,0]$ (show 63), $[8,8,0]$ (show 10)
Associated inertia: $[1]$ (show 6), $[2]$ (show 25), $[3]$ (show 20), $[4]$ (show 6), $[6]$ (show 6), $[8]$ (show 10)
Jump Set: undefined

Fields


Showing 1-50 of 73

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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
3.2.9.32a1.1 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{8} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[8, 8, 0]$ $[2]$ $z^8 + 1$ undefined
3.2.9.32a2.1 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{8} + 3$ not computed $ $not computed$ $ $18$ not computed not computed not computed not computed $[8, 8, 0]$ $[1]$ $z^8 + 2$ undefined
3.2.9.32a2.2 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{8} + 9 x + 3$ $C_3^2\times S_3$ (as 18T17) $54$ $9$ not computed not computed not computed not computed $[8, 8, 0]$ $[1]$ $z^8 + 2$ undefined
3.2.9.32a2.3 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{8} + 12$ not computed $ $not computed$ $ $18$ not computed not computed not computed not computed $[8, 8, 0]$ $[1]$ $z^8 + 2$ undefined
3.2.9.32a2.4 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{8} + 9 x + 12$ $C_3^2\times S_3$ (as 18T17) $54$ $9$ not computed not computed not computed not computed $[8, 8, 0]$ $[1]$ $z^8 + 2$ undefined
3.2.9.32a2.5 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{8} + 21$ $S_3 \times C_3$ (as 18T3) $18$ $18$ not computed not computed not computed not computed $[8, 8, 0]$ $[1]$ $z^8 + 2$ undefined
3.2.9.32a2.6 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{8} + 9 x + 21$ not computed $ $not computed$ $ $9$ not computed not computed not computed not computed $[8, 8, 0]$ $[1]$ $z^8 + 2$ undefined
3.2.9.32a3.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{8} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 8, 0]$ $[8]$ $z^8 + t$ undefined
3.2.9.32a4.1 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{8} + 3$ $C_3^2:F_9$ (as 18T196) $648$ $1$ not computed not computed not computed not computed $[8, 8, 0]$ $[4]$ $z^8 + (t + 1)$ undefined
3.2.9.32a5.1 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{8} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 8, 0]$ $[8]$ $z^8 + 2 t$ undefined
3.2.9.32a6.1 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{8} + 3 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + 2 z^2 + 1$ undefined
3.2.9.32a6.2 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{8} + 3 ( x^{2} + 2 x + 2 )^{6} + 12$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + 2 z^2 + 1$ undefined
3.2.9.32a6.3 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{8} + 3 ( x^{2} + 2 x + 2 )^{6} + 21$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + 2 z^2 + 1$ undefined
3.2.9.32a7.1 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{8} + 3 ( x^{2} + 2 x + 2 )^{6} + 3$ $F_9$ (as 18T28) $72$ $2$ not computed not computed not computed not computed $[8, 6, 0]$ $[4]$ $z^8 + 2 z^2 + 2$ undefined
3.2.9.32a8.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{8} + 3 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + 2 z^2 + t$ undefined
3.2.9.32a8.2 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{8} + 3 ( x^{2} + 2 x + 2 )^{6} + 12$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + 2 z^2 + t$ undefined
3.2.9.32a8.3 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{8} + 3 ( x^{2} + 2 x + 2 )^{6} + 21$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + 2 z^2 + t$ undefined
3.2.9.32a9.1 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{8} + 3 ( x^{2} + 2 x + 2 )^{6} + 3$ $C_3^4:C_{12}$ (as 18T231) $972$ $1$ not computed not computed not computed not computed $[8, 6, 0]$ $[6]$ $z^8 + 2 z^2 + (t + 1)$ undefined
3.2.9.32a10.1 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{8} + 3 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 6, 0]$ $[8]$ $z^8 + 2 z^2 + 2 t$ undefined
3.2.9.32a11.1 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{8} + 6 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $6$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + z^2 + 1$ undefined
3.2.9.32a11.2 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{8} + 6 ( x^{2} + 2 x + 2 )^{6} + 9 x + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + z^2 + 1$ undefined
3.2.9.32a12.1 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{8} + 6 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[8, 6, 0]$ $[4]$ $z^8 + z^2 + 2$ undefined
3.2.9.32a13.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{8} + 6 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + z^2 + t$ undefined
3.2.9.32a13.2 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{8} + 6 ( x^{2} + 2 x + 2 )^{6} + 12$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + z^2 + t$ undefined
3.2.9.32a13.3 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{8} + 6 ( x^{2} + 2 x + 2 )^{6} + 21$ $C_3\wr C_4$ (as 18T123) $324$ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + z^2 + t$ undefined
3.2.9.32a14.1 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{8} + 6 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 6, 0]$ $[6]$ $z^8 + z^2 + (t + 1)$ undefined
3.2.9.32a15.1 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{8} + 6 ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 6, 0]$ $[8]$ $z^8 + z^2 + 2 t$ undefined
3.2.9.32a16.1 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 6, 0]$ $[4]$ $z^8 + 2 t z^2 + 1$ undefined
3.2.9.32a17.1 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 6, 0]$ $[6]$ $z^8 + 2 t z^2 + 2$ undefined
3.2.9.32a18.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 6, 0]$ $[8]$ $z^8 + 2 t z^2 + t$ undefined
3.2.9.32a19.1 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + 2 t z^2 + (t + 1)$ undefined
3.2.9.32a19.2 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 12$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + 2 t z^2 + (t + 1)$ undefined
3.2.9.32a19.3 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 21$ $S_3\times \He_3$ (as 18T77) $162$ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + 2 t z^2 + (t + 1)$ undefined
3.2.9.32a20.1 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + 2 t z^2 + (t + 2)$ undefined
3.2.9.32a20.2 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 12$ $C_3\wr C_4$ (as 18T123) $324$ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + 2 t z^2 + (t + 2)$ undefined
3.2.9.32a20.3 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 6\right) ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 21$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + 2 t z^2 + (t + 2)$ undefined
3.2.9.32a21.1 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + 2 t z^2 + 2 t$ undefined
3.2.9.32a21.2 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 9 x + 3$ $C_3\wr C_4$ (as 18T123) $324$ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + 2 t z^2 + 2 t$ undefined
3.2.9.32a21.3 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 18 x + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + 2 t z^2 + 2 t$ undefined
3.2.9.32a22.1 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 3\right) ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 6, 0]$ $[8]$ $z^8 + 2 t z^2 + (2 t + 1)$ undefined
3.2.9.32a23.1 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + 2 t z^2 + (2 t + 2)$ undefined
3.2.9.32a23.2 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 12$ $C_3^4:C_6$ (as 18T161) $486$ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + 2 t z^2 + (2 t + 2)$ undefined
3.2.9.32a23.3 $( x^{2} + 2 x + 2 )^{9} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + 3 x ( x^{2} + 2 x + 2 )^{6} + 21$ $C_3^4:C_6$ (as 18T161) $486$ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + 2 t z^2 + (2 t + 2)$ undefined
3.2.9.32a24.1 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + (2 t + 2) z^2 + 1$ undefined
3.2.9.32a24.2 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 12$ $C_3^4:C_6$ (as 18T161) $486$ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + (2 t + 2) z^2 + 1$ undefined
3.2.9.32a24.3 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 21$ $C_3^4:C_6$ (as 18T161) $486$ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[3]$ $z^8 + (2 t + 2) z^2 + 1$ undefined
3.2.9.32a25.1 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[8, 6, 0]$ $[4]$ $z^8 + (2 t + 2) z^2 + 2$ undefined
3.2.9.32a26.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + (2 t + 2) z^2 + t$ undefined
3.2.9.32a26.2 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 12$ $C_3^2:C_{12}$ (as 18T44) $108$ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + (2 t + 2) z^2 + t$ undefined
3.2.9.32a26.3 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{8} + \left(6 x + 6\right) ( x^{2} + 2 x + 2 )^{7} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{6} + 21$ $C_3^2:C_{12}$ (as 18T44) $108$ $3$ not computed not computed not computed not computed $[8, 6, 0]$ $[2]$ $z^8 + (2 t + 2) z^2 + t$ undefined
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