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Properties

Label 3.1.6.11a1.7-1.3.13a
Base 3.1.6.11a1.7
Degree \(3\)
e \(3\)
f \(1\)
c \(13\)

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Defining polynomial

$x^{3} + \left(b_{14} \pi^{5} + a_{11} \pi^{4}\right) x^{2} + \left(b_{16} \pi^{6} + b_{13} \pi^{5}\right) x + \pi$

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 $c$: $13$
Absolute Artin slopes: $[\frac{5}{2},\frac{35}{12}]$
Swan slopes: $[\frac{11}{2}]$
Means: $\langle\frac{11}{3}\rangle$
Rams: $(\frac{11}{2})$
Field count: $18$ (complete)
Ambiguity: $1$
Mass: $54$
Absolute Mass: $18$

Diagrams

Varying

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

Indices of inseparability: $[29,18,0]$
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 all 18

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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
3.1.18.46c1.82 $x^{18} + 9 x^{11} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.83 $x^{18} + 9 x^{16} + 9 x^{11} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.84 $x^{18} + 18 x^{16} + 9 x^{11} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.85 $x^{18} + 9 x^{13} + 9 x^{11} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.86 $x^{18} + 9 x^{16} + 9 x^{13} + 9 x^{11} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.87 $x^{18} + 18 x^{16} + 9 x^{13} + 9 x^{11} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.88 $x^{18} + 18 x^{13} + 9 x^{11} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.89 $x^{18} + 9 x^{16} + 18 x^{13} + 9 x^{11} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.90 $x^{18} + 18 x^{16} + 18 x^{13} + 9 x^{11} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.91 $x^{18} + 18 x^{11} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.92 $x^{18} + 9 x^{16} + 18 x^{11} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.93 $x^{18} + 18 x^{16} + 18 x^{11} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.94 $x^{18} + 9 x^{13} + 18 x^{11} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.95 $x^{18} + 9 x^{16} + 9 x^{13} + 18 x^{11} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.96 $x^{18} + 18 x^{16} + 9 x^{13} + 18 x^{11} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.97 $x^{18} + 18 x^{13} + 18 x^{11} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.98 $x^{18} + 9 x^{16} + 18 x^{13} + 18 x^{11} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.99 $x^{18} + 18 x^{16} + 18 x^{13} + 18 x^{11} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
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