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Properties

Label 3.1.6.11a1.7-1.3.19a
Base 3.1.6.11a1.7
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: 3.1.6.11a1.7
Ramification index $e$: $3$
Residue field degree $f$: $1$
Discriminant exponent $c$: $19$
Absolute Artin slopes: $[\frac{5}{2},\frac{41}{12}]$
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: $[35,18,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,7,25]$

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
3.1.18.52a1.730 $x^{18} + 9 x^{17} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.731 $x^{18} + 9 x^{17} + 27 x^{7} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.732 $x^{18} + 9 x^{17} + 54 x^{7} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.733 $x^{18} + 9 x^{17} + 27 x^{5} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.734 $x^{18} + 9 x^{17} + 27 x^{7} + 27 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.735 $x^{18} + 9 x^{17} + 54 x^{7} + 27 x^{5} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.736 $x^{18} + 9 x^{17} + 54 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.737 $x^{18} + 9 x^{17} + 27 x^{7} + 54 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.738 $x^{18} + 9 x^{17} + 54 x^{7} + 54 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.739 $x^{18} + 9 x^{17} + 27 x^{4} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.740 $x^{18} + 9 x^{17} + 27 x^{7} + 27 x^{4} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.741 $x^{18} + 9 x^{17} + 54 x^{7} + 27 x^{4} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.742 $x^{18} + 9 x^{17} + 27 x^{5} + 27 x^{4} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.743 $x^{18} + 9 x^{17} + 27 x^{7} + 27 x^{5} + 27 x^{4} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.744 $x^{18} + 9 x^{17} + 54 x^{7} + 27 x^{5} + 27 x^{4} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.745 $x^{18} + 9 x^{17} + 54 x^{5} + 27 x^{4} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.746 $x^{18} + 9 x^{17} + 27 x^{7} + 54 x^{5} + 27 x^{4} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.747 $x^{18} + 9 x^{17} + 54 x^{7} + 54 x^{5} + 27 x^{4} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.748 $x^{18} + 9 x^{17} + 54 x^{4} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.749 $x^{18} + 9 x^{17} + 27 x^{7} + 54 x^{4} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.750 $x^{18} + 9 x^{17} + 54 x^{7} + 54 x^{4} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.751 $x^{18} + 9 x^{17} + 27 x^{5} + 54 x^{4} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.752 $x^{18} + 9 x^{17} + 27 x^{7} + 27 x^{5} + 54 x^{4} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.753 $x^{18} + 9 x^{17} + 54 x^{7} + 27 x^{5} + 54 x^{4} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.754 $x^{18} + 9 x^{17} + 54 x^{5} + 54 x^{4} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.755 $x^{18} + 9 x^{17} + 27 x^{7} + 54 x^{5} + 54 x^{4} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.756 $x^{18} + 9 x^{17} + 54 x^{7} + 54 x^{5} + 54 x^{4} + 9 x^{3} + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.757 $x^{18} + 9 x^{17} + 9 x^{3} + 27 x + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.758 $x^{18} + 9 x^{17} + 27 x^{7} + 9 x^{3} + 27 x + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.759 $x^{18} + 9 x^{17} + 54 x^{7} + 9 x^{3} + 27 x + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.760 $x^{18} + 9 x^{17} + 27 x^{5} + 9 x^{3} + 27 x + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.761 $x^{18} + 9 x^{17} + 27 x^{7} + 27 x^{5} + 9 x^{3} + 27 x + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.762 $x^{18} + 9 x^{17} + 54 x^{7} + 27 x^{5} + 9 x^{3} + 27 x + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.763 $x^{18} + 9 x^{17} + 54 x^{5} + 9 x^{3} + 27 x + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.764 $x^{18} + 9 x^{17} + 27 x^{7} + 54 x^{5} + 9 x^{3} + 27 x + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.765 $x^{18} + 9 x^{17} + 54 x^{7} + 54 x^{5} + 9 x^{3} + 27 x + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.766 $x^{18} + 9 x^{17} + 27 x^{4} + 9 x^{3} + 27 x + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.767 $x^{18} + 9 x^{17} + 27 x^{7} + 27 x^{4} + 9 x^{3} + 27 x + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.768 $x^{18} + 9 x^{17} + 54 x^{7} + 27 x^{4} + 9 x^{3} + 27 x + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.769 $x^{18} + 9 x^{17} + 27 x^{5} + 27 x^{4} + 9 x^{3} + 27 x + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.770 $x^{18} + 9 x^{17} + 27 x^{7} + 27 x^{5} + 27 x^{4} + 9 x^{3} + 27 x + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.771 $x^{18} + 9 x^{17} + 54 x^{7} + 27 x^{5} + 27 x^{4} + 9 x^{3} + 27 x + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.772 $x^{18} + 9 x^{17} + 54 x^{5} + 27 x^{4} + 9 x^{3} + 27 x + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.773 $x^{18} + 9 x^{17} + 27 x^{7} + 54 x^{5} + 27 x^{4} + 9 x^{3} + 27 x + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.774 $x^{18} + 9 x^{17} + 54 x^{7} + 54 x^{5} + 27 x^{4} + 9 x^{3} + 27 x + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.775 $x^{18} + 9 x^{17} + 54 x^{4} + 9 x^{3} + 27 x + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.776 $x^{18} + 9 x^{17} + 27 x^{7} + 54 x^{4} + 9 x^{3} + 27 x + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.777 $x^{18} + 9 x^{17} + 54 x^{7} + 54 x^{4} + 9 x^{3} + 27 x + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.778 $x^{18} + 9 x^{17} + 27 x^{5} + 54 x^{4} + 9 x^{3} + 27 x + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.52a1.779 $x^{18} + 9 x^{17} + 27 x^{7} + 27 x^{5} + 54 x^{4} + 9 x^{3} + 27 x + 3$ $\He_3^2:\SOPlus(4,2)$ (as 18T725) $52488$ $1$ not computed not computed not computed not computed $[35, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
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