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Properties

Label 3.1.6.7a1.2-1.3.16a
Base 3.1.6.7a1.2
Degree \(3\)
e \(3\)
f \(1\)
c \(16\)

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Defining polynomial

$x^{3} + \left(b_{20} \pi^{7} + b_{17} \pi^{6} + a_{14} \pi^{5}\right) x^{2} + \left(b_{19} \pi^{7} + b_{16} \pi^{6}\right) x + c_{21} \pi^{8} + \pi$

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 $c$: $16$
Absolute Artin slopes: $[\frac{3}{2},\frac{5}{2}]$
Swan slopes: $[7]$
Means: $\langle\frac{14}{3}\rangle$
Rams: $(7)$
Field count: $114$ (complete)
Ambiguity: $3$
Mass: $162$
Absolute Mass: $54$

Diagrams

Varying

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

Indices of inseparability: $[20,6,0]$
Associated inertia: $[1,1,1]$ (show 87), $[1,1,2]$ (show 27)
Jump Set: $[1,5,23]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing 1-50 of 114

Next   displayed columns for results
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
3.1.18.37c1.19 $x^{18} + 3 x^{9} + 3 x^{6} + 18 x^{2} + 3$ $\He_3^2:D_6$ (as 18T534) $8748$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.20 $x^{18} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 18 x^{2} + 3$ $\He_3^2:D_6$ (as 18T534) $8748$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.21 $x^{18} + 3 x^{9} + 18 x^{8} + 3 x^{6} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.22 $x^{18} + 3 x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.23 $x^{18} + 3 x^{9} + 9 x^{8} + 9 x^{7} + 3 x^{6} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.24 $x^{18} + 3 x^{9} + 18 x^{8} + 9 x^{7} + 3 x^{6} + 18 x^{2} + 3$ $\He_3^2:D_6$ (as 18T534) $8748$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.25 $x^{18} + 3 x^{9} + 3 x^{6} + 9 x^{4} + 18 x^{2} + 3$ $C_3^4:S_3^2$ (as 18T409) $2916$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.26 $x^{18} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{4} + 18 x^{2} + 3$ $C_3^4:S_3^2$ (as 18T409) $2916$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.27 $x^{18} + 3 x^{9} + 18 x^{8} + 3 x^{6} + 9 x^{4} + 18 x^{2} + 3$ $C_3^4:S_3^2$ (as 18T409) $2916$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.28 $x^{18} + 3 x^{9} + 9 x^{7} + 3 x^{6} + 9 x^{4} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.29 $x^{18} + 3 x^{9} + 9 x^{8} + 9 x^{7} + 3 x^{6} + 9 x^{4} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.30 $x^{18} + 3 x^{9} + 18 x^{8} + 9 x^{7} + 3 x^{6} + 9 x^{4} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.31 $x^{18} + 3 x^{9} + 3 x^{6} + 18 x^{4} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.32 $x^{18} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 18 x^{4} + 18 x^{2} + 3$ $\He_3^2:D_6$ (as 18T534) $8748$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.33 $x^{18} + 3 x^{9} + 18 x^{8} + 3 x^{6} + 18 x^{4} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.34 $x^{18} + 3 x^{9} + 3 x^{6} + 9 x^{5} + 18 x^{4} + 18 x^{2} + 3$ $\He_3^2:D_6$ (as 18T534) $8748$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.35 $x^{18} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 18 x^{4} + 18 x^{2} + 3$ $\He_3^2:D_6$ (as 18T534) $8748$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.36 $x^{18} + 3 x^{9} + 18 x^{8} + 3 x^{6} + 9 x^{5} + 18 x^{4} + 18 x^{2} + 3$ $\He_3^2:D_6$ (as 18T534) $8748$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.37 $x^{18} + 6 x^{9} + 9 x^{7} + 3 x^{6} + 18 x^{2} + 3$ $\He_3^2:D_6$ (as 18T534) $8748$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.38 $x^{18} + 6 x^{9} + 9 x^{8} + 9 x^{7} + 3 x^{6} + 18 x^{2} + 3$ $\He_3^2:D_6$ (as 18T534) $8748$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.39 $x^{18} + 6 x^{9} + 18 x^{8} + 9 x^{7} + 3 x^{6} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.40 $x^{18} + 6 x^{9} + 9 x^{7} + 3 x^{6} + 9 x^{4} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.41 $x^{18} + 6 x^{9} + 9 x^{8} + 9 x^{7} + 3 x^{6} + 9 x^{4} + 18 x^{2} + 3$ $C_3^4:S_3^2$ (as 18T409) $2916$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.42 $x^{18} + 6 x^{9} + 18 x^{8} + 9 x^{7} + 3 x^{6} + 9 x^{4} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.43 $x^{18} + 6 x^{9} + 3 x^{6} + 9 x^{5} + 18 x^{4} + 18 x^{2} + 3$ $\He_3^2:D_6$ (as 18T534) $8748$ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.44 $x^{18} + 6 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{5} + 18 x^{4} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c1.45 $x^{18} + 6 x^{9} + 18 x^{8} + 3 x^{6} + 9 x^{5} + 18 x^{4} + 18 x^{2} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^6 + 1,z^2 + 1$ $[1, 5, 23]$
3.1.18.37c2.49 $x^{18} + 3 x^{9} + 3 x^{6} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.50 $x^{18} + 12 x^{9} + 3 x^{6} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.51 $x^{18} + 21 x^{9} + 3 x^{6} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.52 $x^{18} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.53 $x^{18} + 12 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.54 $x^{18} + 21 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.55 $x^{18} + 3 x^{9} + 18 x^{8} + 3 x^{6} + 9 x^{2} + 3$ $C_3^4.(C_3\times S_3)$ (as 18T326) $1458$ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.56 $x^{18} + 12 x^{9} + 18 x^{8} + 3 x^{6} + 9 x^{2} + 3$ $C_3^4.(C_3\times S_3)$ (as 18T326) $1458$ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.57 $x^{18} + 21 x^{9} + 18 x^{8} + 3 x^{6} + 9 x^{2} + 3$ $C_3^4.(C_3\times S_3)$ (as 18T326) $1458$ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.58 $x^{18} + 3 x^{9} + 9 x^{7} + 3 x^{6} + 9 x^{2} + 3$ $C_9^2:C_6$ (as 18T158) $486$ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.59 $x^{18} + 12 x^{9} + 9 x^{7} + 3 x^{6} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.60 $x^{18} + 21 x^{9} + 9 x^{7} + 3 x^{6} + 9 x^{2} + 3$ $C_9^2:C_6$ (as 18T158) $486$ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.61 $x^{18} + 3 x^{9} + 9 x^{8} + 9 x^{7} + 3 x^{6} + 9 x^{2} + 3$ $C_9^2:C_6$ (as 18T158) $486$ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.62 $x^{18} + 12 x^{9} + 9 x^{8} + 9 x^{7} + 3 x^{6} + 9 x^{2} + 3$ $C_9^2:C_6$ (as 18T158) $486$ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.63 $x^{18} + 21 x^{9} + 9 x^{8} + 9 x^{7} + 3 x^{6} + 9 x^{2} + 3$ $C_9^2:C_6$ (as 18T158) $486$ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.64 $x^{18} + 3 x^{9} + 18 x^{8} + 9 x^{7} + 3 x^{6} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.65 $x^{18} + 12 x^{9} + 18 x^{8} + 9 x^{7} + 3 x^{6} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.66 $x^{18} + 21 x^{9} + 18 x^{8} + 9 x^{7} + 3 x^{6} + 9 x^{2} + 3$ $C_9^2:C_6$ (as 18T158) $486$ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.67 $x^{18} + 3 x^{9} + 3 x^{6} + 9 x^{4} + 9 x^{2} + 3$ not computed $ $not computed$ $ $9$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.68 $x^{18} + 12 x^{9} + 3 x^{6} + 9 x^{4} + 9 x^{2} + 3$ $C_3^2\wr C_2$ (as 18T79) $162$ $9$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.69 $x^{18} + 21 x^{9} + 3 x^{6} + 9 x^{4} + 9 x^{2} + 3$ $C_3^2\wr C_2$ (as 18T79) $162$ $9$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.70 $x^{18} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 9 x^{4} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
3.1.18.37c2.71 $x^{18} + 3 x^{9} + 18 x^{8} + 3 x^{6} + 9 x^{4} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[20, 6, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z^2 + 2$ $[1, 5, 23]$
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