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Properties

Label 3.1.6.7a1.2-1.3.19a
Base 3.1.6.7a1.2
Degree \(3\)
e \(3\)
f \(1\)
c \(19\)

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Defining polynomial

$x^{3} + \left(b_{23} \pi^{8} + b_{20} \pi^{7} + a_{17} \pi^{6}\right) x^{2} + \left(b_{25} \pi^{9} + b_{22} \pi^{8} + b_{19} \pi^{7}\right) x + \pi$

Invariants

Residue field characteristic: $3$
Degree: $3$
Base field: 3.1.6.7a1.2
Ramification index $e$: $3$
Residue field degree $f$: $1$
Discriminant exponent $c$: $19$
Absolute Artin slopes: $[\frac{3}{2},\frac{11}{4}]$
Swan slopes: $[\frac{17}{2}]$
Means: $\langle\frac{17}{3}\rangle$
Rams: $(\frac{17}{2})$
Field count: $162$ (complete)
Ambiguity: $1$
Mass: $486$
Absolute Mass: $162$

Diagrams

Varying

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

Indices of inseparability: $[23,6,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,5,23]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing 1-50 of 162

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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
3.1.18.40b1.82 $x^{18} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.83 $x^{18} + 9 x^{13} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.84 $x^{18} + 18 x^{13} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.85 $x^{18} + 9 x^{11} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.86 $x^{18} + 9 x^{13} + 9 x^{11} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.87 $x^{18} + 18 x^{13} + 9 x^{11} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.88 $x^{18} + 18 x^{11} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.89 $x^{18} + 9 x^{13} + 18 x^{11} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.90 $x^{18} + 18 x^{13} + 18 x^{11} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.91 $x^{18} + 9 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.92 $x^{18} + 9 x^{13} + 9 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.93 $x^{18} + 18 x^{13} + 9 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.94 $x^{18} + 9 x^{11} + 9 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.95 $x^{18} + 9 x^{13} + 9 x^{11} + 9 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.96 $x^{18} + 18 x^{13} + 9 x^{11} + 9 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.97 $x^{18} + 18 x^{11} + 9 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.98 $x^{18} + 9 x^{13} + 18 x^{11} + 9 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.99 $x^{18} + 18 x^{13} + 18 x^{11} + 9 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.100 $x^{18} + 18 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.101 $x^{18} + 9 x^{13} + 18 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.102 $x^{18} + 18 x^{13} + 18 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.103 $x^{18} + 9 x^{11} + 18 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.104 $x^{18} + 9 x^{13} + 9 x^{11} + 18 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.105 $x^{18} + 18 x^{13} + 9 x^{11} + 18 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.106 $x^{18} + 18 x^{11} + 18 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.107 $x^{18} + 9 x^{13} + 18 x^{11} + 18 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.108 $x^{18} + 18 x^{13} + 18 x^{11} + 18 x^{10} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.109 $x^{18} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.110 $x^{18} + 9 x^{13} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.111 $x^{18} + 18 x^{13} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.112 $x^{18} + 9 x^{11} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.113 $x^{18} + 9 x^{13} + 9 x^{11} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.114 $x^{18} + 18 x^{13} + 9 x^{11} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.115 $x^{18} + 18 x^{11} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.116 $x^{18} + 9 x^{13} + 18 x^{11} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.117 $x^{18} + 18 x^{13} + 18 x^{11} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.118 $x^{18} + 9 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.119 $x^{18} + 9 x^{13} + 9 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.120 $x^{18} + 18 x^{13} + 9 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.121 $x^{18} + 9 x^{11} + 9 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.122 $x^{18} + 9 x^{13} + 9 x^{11} + 9 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.123 $x^{18} + 18 x^{13} + 9 x^{11} + 9 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.124 $x^{18} + 18 x^{11} + 9 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.125 $x^{18} + 9 x^{13} + 18 x^{11} + 9 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.126 $x^{18} + 18 x^{13} + 18 x^{11} + 9 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.127 $x^{18} + 18 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.128 $x^{18} + 9 x^{13} + 18 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.129 $x^{18} + 18 x^{13} + 18 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.130 $x^{18} + 9 x^{11} + 18 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
3.1.18.40b1.131 $x^{18} + 9 x^{13} + 9 x^{11} + 18 x^{10} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[23, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 5, 23]$
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