Properties

Label 3.3.3.9a
Base 3.1.1.0a1.1
Degree \(9\)
e \(3\)
f \(3\)
c \(9\)

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Defining polynomial over unramified subextension

$x^{3} + 3a_{1} x + 3$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}$
Ramification index $e$: $3$
Residue field degree $f$: $3$
Discriminant exponent $c$: $9$
Artin slopes: $[\frac{3}{2}]$
Swan slopes: $[\frac{1}{2}]$
Means: $\langle\frac{1}{3}\rangle$
Rams: $(\frac{1}{2})$
Field count: $10$ (complete)
Ambiguity: $3$
Mass: $26$
Absolute Mass: $26/3$

Diagrams

Varying

Indices of inseparability: $[1,0]$
Associated inertia: $[1]$
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 10

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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
3.3.3.9a1.1 1 $( x^{3} + 2 x + 1 )^{3} + 6 ( x^{3} + 2 x + 1 ) + 3$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[\frac{3}{2}]_{2}^{3}$ $[\frac{1}{2}]_{2}^{3}$ $[\ ]_{2}$ $[\ ]_{2}$ $[1, 0]$ $[1]$ $z + 1$ undefined
3.3.3.9a2.1 1 $( x^{3} + 2 x + 1 )^{3} + 3 ( x^{3} + 2 x + 1 ) + 3$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[\frac{3}{2}]_{2}^{3}$ $[\frac{1}{2}]_{2}^{3}$ $[\ ]_{2}$ $[\ ]_{2}$ $[1, 0]$ $[1]$ $z + 2$ undefined
3.3.3.9a3.1 1 $( x^{3} + 2 x + 1 )^{3} + 6 x ( x^{3} + 2 x + 1 ) + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[1, 0]$ $[1]$ $z + t$ undefined
3.3.3.9a4.1 1 $( x^{3} + 2 x + 1 )^{3} + 3 x ( x^{3} + 2 x + 1 ) + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[1, 0]$ $[1]$ $z + 2 t$ undefined
3.3.3.9a5.1 1 $( x^{3} + 2 x + 1 )^{3} + 6 x^{2} ( x^{3} + 2 x + 1 ) + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[1, 0]$ $[1]$ $z + t^2$ undefined
3.3.3.9a6.1 1 $( x^{3} + 2 x + 1 )^{3} + \left(6 x^{2} + 3 x\right) ( x^{3} + 2 x + 1 ) + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[1, 0]$ $[1]$ $z + (t^2 + 2 t)$ undefined
3.3.3.9a7.1 1 $( x^{3} + 2 x + 1 )^{3} + \left(6 x^{2} + 3 x + 3\right) ( x^{3} + 2 x + 1 ) + 3$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[\frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[1, 0]$ $[1]$ $z + (t^2 + 2 t + 2)$ undefined
3.3.3.9a8.1 1 $( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 3\right) ( x^{3} + 2 x + 1 ) + 3$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[\frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[1, 0]$ $[1]$ $z + (2 t^2 + 2)$ undefined
3.3.3.9a9.1 1 $( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 6 x\right) ( x^{3} + 2 x + 1 ) + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[1, 0]$ $[1]$ $z + (2 t^2 + t)$ undefined
3.3.3.9a10.1 1 $( x^{3} + 2 x + 1 )^{3} + \left(3 x^{2} + 6 x + 3\right) ( x^{3} + 2 x + 1 ) + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[1, 0]$ $[1]$ $z + (2 t^2 + t + 2)$ undefined
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