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Properties

Label 3.1.6.10a1.1-1.3.20a
Base 3.1.6.10a1.1
Degree \(3\)
e \(3\)
f \(1\)
c \(20\)

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Defining polynomial

$x^{3} + \left(b_{26} \pi^{9} + b_{23} \pi^{8} + b_{20} \pi^{7}\right) x^{2} + \left(b_{25} \pi^{9} + b_{22} \pi^{8} + b_{19} \pi^{7}\right) x + c_{27} \pi^{10} + \pi$

Invariants

Residue field characteristic: $3$
Degree: $3$
Base field: 3.1.6.10a1.1
Ramification index $e$: $3$
Residue field degree $f$: $1$
Discriminant exponent $c$: $20$
Absolute Artin slopes: $[\frac{9}{4},\frac{10}{3}]$
Swan slopes: $[9]$
Means: $\langle6\rangle$
Rams: $(9)$
Field count: $2187$ (complete)
Ambiguity: $3$
Mass: $729$
Absolute Mass: $729$

Diagrams

Varying

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

Indices of inseparability: $[33,15,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,7,25]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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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
3.1.18.50a2.1 $x^{18} + 3 x^{15} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.2 $x^{18} + 3 x^{15} + 27 x^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.3 $x^{18} + 3 x^{15} + 54 x^{6} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.4 $x^{18} + 3 x^{15} + 27 x^{5} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.5 $x^{18} + 3 x^{15} + 27 x^{6} + 27 x^{5} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.6 $x^{18} + 3 x^{15} + 54 x^{6} + 27 x^{5} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.7 $x^{18} + 3 x^{15} + 54 x^{5} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.8 $x^{18} + 3 x^{15} + 27 x^{6} + 54 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.9 $x^{18} + 3 x^{15} + 54 x^{6} + 54 x^{5} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.10 $x^{18} + 3 x^{15} + 27 x^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.11 $x^{18} + 3 x^{15} + 27 x^{6} + 27 x^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.12 $x^{18} + 3 x^{15} + 54 x^{6} + 27 x^{4} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.13 $x^{18} + 3 x^{15} + 27 x^{5} + 27 x^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.14 $x^{18} + 3 x^{15} + 27 x^{6} + 27 x^{5} + 27 x^{4} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.15 $x^{18} + 3 x^{15} + 54 x^{6} + 27 x^{5} + 27 x^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.16 $x^{18} + 3 x^{15} + 54 x^{5} + 27 x^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.17 $x^{18} + 3 x^{15} + 27 x^{6} + 54 x^{5} + 27 x^{4} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.18 $x^{18} + 3 x^{15} + 54 x^{6} + 54 x^{5} + 27 x^{4} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.19 $x^{18} + 3 x^{15} + 27 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.20 $x^{18} + 3 x^{15} + 27 x^{6} + 27 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.21 $x^{18} + 3 x^{15} + 54 x^{6} + 27 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.22 $x^{18} + 3 x^{15} + 27 x^{5} + 27 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.23 $x^{18} + 3 x^{15} + 27 x^{6} + 27 x^{5} + 27 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.24 $x^{18} + 3 x^{15} + 54 x^{6} + 27 x^{5} + 27 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.25 $x^{18} + 3 x^{15} + 54 x^{5} + 27 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.26 $x^{18} + 3 x^{15} + 27 x^{6} + 54 x^{5} + 27 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.27 $x^{18} + 3 x^{15} + 54 x^{6} + 54 x^{5} + 27 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.28 $x^{18} + 3 x^{15} + 27 x^{4} + 27 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.29 $x^{18} + 3 x^{15} + 27 x^{6} + 27 x^{4} + 27 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.30 $x^{18} + 3 x^{15} + 54 x^{6} + 27 x^{4} + 27 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.31 $x^{18} + 3 x^{15} + 27 x^{5} + 27 x^{4} + 27 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.32 $x^{18} + 3 x^{15} + 27 x^{6} + 27 x^{5} + 27 x^{4} + 27 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.33 $x^{18} + 3 x^{15} + 54 x^{6} + 27 x^{5} + 27 x^{4} + 27 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.34 $x^{18} + 3 x^{15} + 54 x^{5} + 27 x^{4} + 27 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.35 $x^{18} + 3 x^{15} + 27 x^{6} + 54 x^{5} + 27 x^{4} + 27 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.36 $x^{18} + 3 x^{15} + 54 x^{6} + 54 x^{5} + 27 x^{4} + 27 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.37 $x^{18} + 3 x^{15} + 54 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.38 $x^{18} + 3 x^{15} + 27 x^{6} + 54 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.39 $x^{18} + 3 x^{15} + 54 x^{6} + 54 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.40 $x^{18} + 3 x^{15} + 27 x^{5} + 54 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.41 $x^{18} + 3 x^{15} + 27 x^{6} + 27 x^{5} + 54 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.42 $x^{18} + 3 x^{15} + 54 x^{6} + 27 x^{5} + 54 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.43 $x^{18} + 3 x^{15} + 54 x^{5} + 54 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.44 $x^{18} + 3 x^{15} + 27 x^{6} + 54 x^{5} + 54 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.45 $x^{18} + 3 x^{15} + 54 x^{6} + 54 x^{5} + 54 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.46 $x^{18} + 3 x^{15} + 27 x^{4} + 54 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.47 $x^{18} + 3 x^{15} + 27 x^{6} + 27 x^{4} + 54 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.48 $x^{18} + 3 x^{15} + 54 x^{6} + 27 x^{4} + 54 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.49 $x^{18} + 3 x^{15} + 27 x^{5} + 27 x^{4} + 54 x^{2} + 3$ $C_3^5:\SOPlus(4,2)$ (as 18T614) $17496$ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.50a2.50 $x^{18} + 3 x^{15} + 27 x^{6} + 27 x^{5} + 27 x^{4} + 54 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[33, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
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