Properties

Label 3.1.9.18a
Base 3.1.1.0a1.1
Degree \(9\)
e \(9\)
f \(1\)
c \(18\)

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Defining polynomial

$x^{9} + 9 b_{11} x^{2} + 9 a_{10} x + 3$

Invariants

Residue field characteristic: $3$
Degree: $9$
Base field: $\Q_{3}$
Ramification index $e$: $9$
Residue field degree $f$: $1$
Discriminant exponent $c$: $18$
Artin slopes: $[\frac{9}{4},\frac{9}{4}]$
Swan slopes: $[\frac{5}{4},\frac{5}{4}]$
Means: $\langle\frac{5}{6},\frac{10}{9}\rangle$
Rams: $(\frac{5}{4},\frac{5}{4})$
Field count: $6$ (complete)
Ambiguity: $1$
Mass: $6$
Absolute Mass: $6$

Diagrams

Varying

Indices of inseparability: $[10,9,0]$
Associated inertia: $[1]$ (show 3), $[2]$ (show 3)
Jump Set: undefined

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
3.1.9.18a1.1 3 $x^{9} + 9 x + 3$ $C_3^2:Q_8$ (as 9T14) $72$ $1$ $[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]_{4}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[10, 9, 0]$ $[2]$ $z^2 + 1$ undefined
3.1.9.18a1.2 3 $x^{9} + 9 x^{2} + 9 x + 3$ $C_3^2:Q_8$ (as 9T14) $72$ $1$ $[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]_{4}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[10, 9, 0]$ $[2]$ $z^2 + 1$ undefined
3.1.9.18a1.3 3 $x^{9} + 18 x^{2} + 9 x + 3$ $C_3^2:Q_8$ (as 9T14) $72$ $1$ $[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]_{4}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[10, 9, 0]$ $[2]$ $z^2 + 1$ undefined
3.1.9.18a2.1 3 $x^{9} + 18 x + 3$ $S_3^2:C_2$ (as 9T16) $72$ $1$ $[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]_{4}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[10, 9, 0]$ $[1]$ $z^2 + 2$ undefined
3.1.9.18a2.2 3 $x^{9} + 9 x^{2} + 18 x + 3$ $S_3^2:C_2$ (as 9T16) $72$ $1$ $[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]_{4}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[10, 9, 0]$ $[1]$ $z^2 + 2$ undefined
3.1.9.18a2.3 3 $x^{9} + 18 x^{2} + 18 x + 3$ $S_3^2:C_2$ (as 9T16) $72$ $1$ $[\frac{9}{4}, \frac{9}{4}]_{4}^{2}$ $[\frac{5}{4},\frac{5}{4}]_{4}^{2}$ $[\ ]^{2}_{4}$ $[\ ]^{2}_{4}$ $[10, 9, 0]$ $[1]$ $z^2 + 2$ undefined
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