Properties

Label 3.3.3.15a
Base 3.1.1.0a1.1
Degree \(9\)
e \(3\)
f \(3\)
c \(15\)

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Defining polynomial over unramified subextension

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

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}$
Ramification index $e$: $3$
Residue field degree $f$: $3$
Discriminant exponent $c$: $15$
Artin slopes: $[\frac{5}{2}]$
Swan slopes: $[\frac{3}{2}]$
Means: $\langle1\rangle$
Rams: $(\frac{3}{2})$
Field count: $11$ (complete)
Ambiguity: $3$
Mass: $27$
Absolute Mass: $9$

Diagrams

Varying

Indices of inseparability: $[3,0]$
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 11

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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
3.3.3.15a1.1 3 $( x^{3} + 2 x + 1 )^{3} + 3$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[\frac{5}{2}]_{2}^{3}$ $[\frac{3}{2}]_{2}^{3}$ $[\ ]_{2}$ $[\ ]_{2}$ $[3, 0]$ $[1]$ $z + 2$ undefined
3.3.3.15a1.2 6 $( x^{3} + 2 x + 1 )^{3} + 9 x^{2} ( x^{3} + 2 x + 1 ) + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{3}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[3, 0]$ $[1]$ $z + 2$ undefined
3.3.3.15a1.3 6 $( x^{3} + 2 x + 1 )^{3} + 18 x^{2} ( x^{3} + 2 x + 1 ) + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{3}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[3, 0]$ $[1]$ $z + 2$ undefined
3.3.3.15a1.4 2 $( x^{3} + 2 x + 1 )^{3} + 9 x ( x^{3} + 2 x + 1 ) + 3$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[\frac{3}{2}, \frac{5}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{3}{2}]_{2}^{3}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[3, 0]$ $[1]$ $z + 2$ undefined
3.3.3.15a1.5 2 $( x^{3} + 2 x + 1 )^{3} + 18 x ( x^{3} + 2 x + 1 ) + 3$ $C_3^2 : S_3 $ (as 9T13) $54$ $1$ $[\frac{3}{2}, \frac{5}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{3}{2}]_{2}^{3}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[3, 0]$ $[1]$ $z + 2$ undefined
3.3.3.15a1.6 3 $( x^{3} + 2 x + 1 )^{3} + 9 ( x^{3} + 2 x + 1 ) + 3$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[\frac{5}{2}]_{2}^{3}$ $[\frac{3}{2}]_{2}^{3}$ $[\ ]_{2}$ $[\ ]_{2}$ $[3, 0]$ $[1]$ $z + 2$ undefined
3.3.3.15a1.7 6 $( x^{3} + 2 x + 1 )^{3} + \left(9 x^{2} + 9\right) ( x^{3} + 2 x + 1 ) + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{3}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[3, 0]$ $[1]$ $z + 2$ undefined
3.3.3.15a1.8 6 $( x^{3} + 2 x + 1 )^{3} + \left(18 x^{2} + 9\right) ( x^{3} + 2 x + 1 ) + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{3}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[3, 0]$ $[1]$ $z + 2$ undefined
3.3.3.15a1.9 3 $( x^{3} + 2 x + 1 )^{3} + 18 ( x^{3} + 2 x + 1 ) + 3$ $S_3\times C_3$ (as 9T4) $18$ $3$ $[\frac{5}{2}]_{2}^{3}$ $[\frac{3}{2}]_{2}^{3}$ $[\ ]_{2}$ $[\ ]_{2}$ $[3, 0]$ $[1]$ $z + 2$ undefined
3.3.3.15a1.10 6 $( x^{3} + 2 x + 1 )^{3} + \left(9 x^{2} + 18\right) ( x^{3} + 2 x + 1 ) + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{3}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[3, 0]$ $[1]$ $z + 2$ undefined
3.3.3.15a1.11 6 $( x^{3} + 2 x + 1 )^{3} + \left(18 x^{2} + 18\right) ( x^{3} + 2 x + 1 ) + 3$ $(C_3^3:C_3):C_2$ (as 9T22) $162$ $1$ $[\frac{3}{2}, \frac{3}{2}, \frac{5}{2}]_{2}^{3}$ $[\frac{1}{2},\frac{1}{2},\frac{3}{2}]_{2}^{3}$ $[\frac{3}{2},\frac{3}{2}]_{2}$ $[\frac{1}{2},\frac{1}{2}]_{2}$ $[3, 0]$ $[1]$ $z + 2$ undefined
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