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Properties

Label 3.1.6.9a1.11-1.3.10a
Base 3.1.6.9a1.11
Degree \(3\)
e \(3\)
f \(1\)
c \(10\)

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Defining polynomial

$x^{3} + \left(b_{11} \pi^{4} + a_{8} \pi^{3}\right) x^{2} + b_{10} \pi^{4} x + c_{12} \pi^{5} + \pi$

Invariants

Residue field characteristic: $3$
Degree: $3$
Base field: 3.1.6.9a1.11
Ramification index $e$: $3$
Residue field degree $f$: $1$
Discriminant exponent $c$: $10$
Absolute Artin slopes: $[2,\frac{7}{3}]$
Swan slopes: $[4]$
Means: $\langle\frac{8}{3}\rangle$
Rams: $(4)$
Field count: $12$ (complete)
Ambiguity: $3$
Mass: $18$
Absolute Mass: $6$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.

Indices of inseparability: $[20,12,0]$
Associated inertia: $[1,1,1]$ (show 9), $[1,1,2]$ (show 3)
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 12

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Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
3.1.18.37d1.24 $x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{2} + 15$ $C_3^4:S_3^2$ (as 18T409) $2916$ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.25 $x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 15$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d1.29 $x^{18} + 6 x^{15} + 3 x^{12} + 9 x^{5} + 18 x^{2} + 15$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ undefined
3.1.18.37d2.70 $x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{2} + 15$ $\He_3^2:C_6$ (as 18T455) $4374$ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.37d2.71 $x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{6} + 9 x^{2} + 15$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.37d2.72 $x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{6} + 9 x^{2} + 15$ $\He_3^2:C_6$ (as 18T455) $4374$ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.37d2.73 $x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{5} + 9 x^{2} + 15$ $\He_3^2:C_6$ (as 18T455) $4374$ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.37d2.74 $x^{18} + 3 x^{15} + 3 x^{12} + 9 x^{6} + 9 x^{5} + 9 x^{2} + 15$ $\He_3^2:C_6$ (as 18T455) $4374$ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.37d2.75 $x^{18} + 3 x^{15} + 3 x^{12} + 18 x^{6} + 9 x^{5} + 9 x^{2} + 15$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.37d2.85 $x^{18} + 6 x^{15} + 3 x^{12} + 9 x^{5} + 9 x^{2} + 15$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.37d2.86 $x^{18} + 6 x^{15} + 3 x^{12} + 9 x^{6} + 9 x^{5} + 9 x^{2} + 15$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
3.1.18.37d2.87 $x^{18} + 6 x^{15} + 3 x^{12} + 18 x^{6} + 9 x^{5} + 9 x^{2} + 15$ $\He_3^2:C_6$ (as 18T455) $4374$ $3$ not computed not computed not computed not computed $[20, 12, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ undefined
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