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Properties

Label 3.1.6.6a
Base 3.1.1.0a1.1
Degree \(6\)
e \(6\)
f \(1\)
c \(6\)

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Defining polynomial

$x^{6} + 3 a_{1} x + 3 d_{0}$

Invariants

Residue field characteristic: $3$
Degree: $6$
Base field: $\Q_{3}$
Ramification index $e$: $6$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Artin slopes: $[\frac{5}{4}]$
Swan slopes: $[\frac{1}{4}]$
Means: $\langle\frac{1}{6}\rangle$
Rams: $(\frac{1}{2})$
Field count: $2$ (complete)
Ambiguity: $2$
Mass: $2$
Absolute Mass: $2$

Diagrams

Varying

Indices of inseparability: $[1,0]$
Associated inertia: $[1,1]$
Jump Set: undefined (show 1), $[1,4]$ (show 1)

Galois groups and Hidden Artin slopes

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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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
3.1.6.6a1.1 2 $x^{6} + 3 x + 3$ $C_3^2:D_4$ (as 6T13) $72$ $1$ $[\frac{5}{4}, \frac{5}{4}]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4}]_{4}^{2}$ $[\frac{5}{4}]^{2}_{2}$ $[\frac{1}{4}]^{2}_{2}$ $[1, 0]$ $[1, 1]$ $z^3 + 2,2 z + 2$ $[1, 4]$
3.1.6.6a1.2 2 $x^{6} + 3 x + 6$ $C_3^2:D_4$ (as 6T13) $72$ $1$ $[\frac{5}{4}, \frac{5}{4}]_{4}^{2}$ $[\frac{1}{4},\frac{1}{4}]_{4}^{2}$ $[\frac{5}{4}]^{2}_{2}$ $[\frac{1}{4}]^{2}_{2}$ $[1, 0]$ $[1, 1]$ $z^3 + 2,2 z + 1$ undefined
  displayed columns for results