Properties

Label 3.3.3.12a
Base 3.1.1.0a1.1
Degree \(9\)
e \(3\)
f \(3\)
c \(12\)

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Defining polynomial over unramified subextension

$x^{3} + 3 a_{2} x^{2} + 9 c_{3} + 3$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}$
Ramification index $e$: $3$
Residue field degree $f$: $3$
Discriminant exponent $c$: $12$
Artin slopes: $[2]$
Swan slopes: $[1]$
Means: $\langle\frac{2}{3}\rangle$
Rams: $(1)$
Field count: $20$ (complete)
Ambiguity: $9$
Mass: $26$
Absolute Mass: $26/3$

Diagrams

Varying

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

Galois groups and Hidden Artin slopes

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Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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Showing all 20

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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
3.3.3.12a1.1 1 $( x^{3} + 2 x + 1 )^{3} + 3 ( x^{3} + 2 x + 1 )^{2} + 3$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[2]^{6}$ $[1]^{6}$ $[\ ]^{2}$ $[\ ]^{2}$ $[2, 0]$ $[2]$ $z^2 + 1$ undefined
3.3.3.12a2.1 1 $( x^{3} + 2 x + 1 )^{3} + 6 ( x^{3} + 2 x + 1 )^{2} + 3$ $C_3^2$ (as 9T2) $9$ $9$ $[2]^{3}$ $[1]^{3}$ $[\ ]$ $[\ ]$ $[2, 0]$ $[1]$ $z^2 + 2$ undefined
3.3.3.12a2.2 2 $( x^{3} + 2 x + 1 )^{3} + 6 ( x^{3} + 2 x + 1 )^{2} + 9 x^{2} + 3$ $C_9$ (as 9T1) $9$ $9$ $[2]^{3}$ $[1]^{3}$ $[\ ]$ $[\ ]$ $[2, 0]$ $[1]$ $z^2 + 2$ undefined
3.3.3.12a2.3 2 $( x^{3} + 2 x + 1 )^{3} + 6 ( x^{3} + 2 x + 1 )^{2} + 18 x^{2} + 3$ $C_9$ (as 9T1) $9$ $9$ $[2]^{3}$ $[1]^{3}$ $[\ ]$ $[\ ]$ $[2, 0]$ $[1]$ $z^2 + 2$ undefined
3.3.3.12a3.1 3 $( x^{3} + 2 x + 1 )^{3} + 3 x ( x^{3} + 2 x + 1 )^{2} + 3$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, 2]^{3}$ $[1,1,1]^{3}$ $[2,2]$ $[1,1]$ $[2, 0]$ $[1]$ $z^2 + t$ undefined
3.3.3.12a3.2 3 $( x^{3} + 2 x + 1 )^{3} + 3 x ( x^{3} + 2 x + 1 )^{2} + 12$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, 2]^{3}$ $[1,1,1]^{3}$ $[2,2]$ $[1,1]$ $[2, 0]$ $[1]$ $z^2 + t$ undefined
3.3.3.12a3.3 3 $( x^{3} + 2 x + 1 )^{3} + 3 x ( x^{3} + 2 x + 1 )^{2} + 21$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, 2]^{3}$ $[1,1,1]^{3}$ $[2,2]$ $[1,1]$ $[2, 0]$ $[1]$ $z^2 + t$ undefined
3.3.3.12a4.1 1 $( x^{3} + 2 x + 1 )^{3} + 6 x ( x^{3} + 2 x + 1 )^{2} + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[2, 2, 2]^{6}$ $[1,1,1]^{6}$ $[2,2]^{2}$ $[1,1]^{2}$ $[2, 0]$ $[2]$ $z^2 + 2 t$ undefined
3.3.3.12a5.1 1 $( x^{3} + 2 x + 1 )^{3} + 3 x^{2} ( x^{3} + 2 x + 1 )^{2} + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[2, 2, 2]^{6}$ $[1,1,1]^{6}$ $[2,2]^{2}$ $[1,1]^{2}$ $[2, 0]$ $[2]$ $z^2 + t^2$ undefined
3.3.3.12a6.1 1 $( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 6 x\right) ( x^{3} + 2 x + 1 )^{2} + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[2, 2, 2]^{6}$ $[1,1,1]^{6}$ $[2,2]^{2}$ $[1,1]^{2}$ $[2, 0]$ $[2]$ $z^2 + (t^2 + 2 t)$ undefined
3.3.3.12a7.1 2 $( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 6 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3$ $C_9:C_3$ (as 9T6) $27$ $3$ $[2, 2]^{3}$ $[1,1]^{3}$ $[2]$ $[1]$ $[2, 0]$ $[1]$ $z^2 + (t^2 + 2 t + 2)$ undefined
3.3.3.12a7.2 1 $( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 6 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 9 x + 3$ $C_3^2:C_3$ (as 9T7) $27$ $3$ $[2, 2]^{3}$ $[1,1]^{3}$ $[2]$ $[1]$ $[2, 0]$ $[1]$ $z^2 + (t^2 + 2 t + 2)$ undefined
3.3.3.12a7.3 2 $( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 6 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 18 x + 3$ $C_9:C_3$ (as 9T6) $27$ $3$ $[2, 2]^{3}$ $[1,1]^{3}$ $[2]$ $[1]$ $[2, 0]$ $[1]$ $z^2 + (t^2 + 2 t + 2)$ undefined
3.3.3.12a8.1 1 $( x^{3} + 2 x + 1 )^{3} + \left(6 x^{2} + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[2, 2]^{6}$ $[1,1]^{6}$ $[2]^{2}$ $[1]^{2}$ $[2, 0]$ $[2]$ $z^2 + (2 t^2 + 2)$ undefined
3.3.3.12a9.1 3 $( x^{3} + 2 x + 1 )^{3} + \left(6 x^{2} + 3 x\right) ( x^{3} + 2 x + 1 )^{2} + 3$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, 2]^{3}$ $[1,1,1]^{3}$ $[2,2]$ $[1,1]$ $[2, 0]$ $[1]$ $z^2 + (2 t^2 + t)$ undefined
3.3.3.12a9.2 3 $( x^{3} + 2 x + 1 )^{3} + \left(6 x^{2} + 3 x\right) ( x^{3} + 2 x + 1 )^{2} + 12$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, 2]^{3}$ $[1,1,1]^{3}$ $[2,2]$ $[1,1]$ $[2, 0]$ $[1]$ $z^2 + (2 t^2 + t)$ undefined
3.3.3.12a9.3 3 $( x^{3} + 2 x + 1 )^{3} + \left(6 x^{2} + 3 x\right) ( x^{3} + 2 x + 1 )^{2} + 21$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, 2]^{3}$ $[1,1,1]^{3}$ $[2,2]$ $[1,1]$ $[2, 0]$ $[1]$ $z^2 + (2 t^2 + t)$ undefined
3.3.3.12a10.1 3 $( x^{3} + 2 x + 1 )^{3} + \left(6 x^{2} + 3 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 3$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, 2]^{3}$ $[1,1,1]^{3}$ $[2,2]$ $[1,1]$ $[2, 0]$ $[1]$ $z^2 + (2 t^2 + t + 2)$ undefined
3.3.3.12a10.2 3 $( x^{3} + 2 x + 1 )^{3} + \left(6 x^{2} + 3 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 12$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, 2]^{3}$ $[1,1,1]^{3}$ $[2,2]$ $[1,1]$ $[2, 0]$ $[1]$ $z^2 + (2 t^2 + t + 2)$ undefined
3.3.3.12a10.3 3 $( x^{3} + 2 x + 1 )^{3} + \left(6 x^{2} + 3 x + 6\right) ( x^{3} + 2 x + 1 )^{2} + 21$ $C_3 \wr C_3 $ (as 9T17) $81$ $3$ $[2, 2, 2]^{3}$ $[1,1,1]^{3}$ $[2,2]$ $[1,1]$ $[2, 0]$ $[1]$ $z^2 + (2 t^2 + t + 2)$ undefined
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