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Properties

Label 3.1.18.40d
Base 3.1.1.0a1.1
Degree \(18\)
e \(18\)
f \(1\)
c \(40\)

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Defining polynomial

$x^{18} + 3 a_{15} x^{15} + 9 c_{27} x^{9} + 9 b_{26} x^{8} + 9 b_{25} x^{7} + 9 a_{23} x^{5} + 9 b_{21} x^{3} + 3 d_{0}$

Invariants

Residue field characteristic: $3$
Degree: $18$
Base field: $\Q_{3}$
Ramification index $e$: $18$
Residue field degree $f$: $1$
Discriminant exponent $c$: $40$
Artin slopes: $[\frac{9}{4},\frac{5}{2}]$
Swan slopes: $[\frac{5}{4},\frac{3}{2}]$
Means: $\langle\frac{5}{6},\frac{23}{18}\rangle$
Rams: $(\frac{5}{2},4)$
Field count: $216$ (complete)
Ambiguity: $6$
Mass: $108$
Absolute Mass: $108$

Diagrams

Varying

Indices of inseparability: $[23,15,0]$
Associated inertia: $[1,1,1]$ (show 162), $[1,1,2]$ (show 54)
Jump Set: undefined (show 108), $[1,7,25]$ (show 108)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing 1-50 of 108

Next   displayed columns for results
Label Packet size Polynomial Galois group Galois degree $\#\Aut(K/\Q_p)$ Artin slope content Swan slope content Hidden Artin slopes Hidden Swan slopes Ind. of Insep. Assoc. Inertia Resid. Poly Jump Set
3.1.18.40d1.1 $x^{18} + 3 x^{15} + 18 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 1,z^2 + 1$ $[1, 7, 25]$
3.1.18.40d1.2 $x^{18} + 3 x^{15} + 9 x^{8} + 18 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 1,z^2 + 1$ $[1, 7, 25]$
3.1.18.40d1.3 $x^{18} + 3 x^{15} + 18 x^{8} + 18 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 1,z^2 + 1$ $[1, 7, 25]$
3.1.18.40d1.4 $x^{18} + 3 x^{15} + 9 x^{7} + 18 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 1,z^2 + 1$ $[1, 7, 25]$
3.1.18.40d1.5 $x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{7} + 18 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 1,z^2 + 1$ $[1, 7, 25]$
3.1.18.40d1.6 $x^{18} + 3 x^{15} + 18 x^{8} + 9 x^{7} + 18 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 1,z^2 + 1$ $[1, 7, 25]$
3.1.18.40d1.7 $x^{18} + 3 x^{15} + 18 x^{7} + 18 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 1,z^2 + 1$ $[1, 7, 25]$
3.1.18.40d1.8 $x^{18} + 3 x^{15} + 9 x^{8} + 18 x^{7} + 18 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 1,z^2 + 1$ $[1, 7, 25]$
3.1.18.40d1.9 $x^{18} + 3 x^{15} + 18 x^{8} + 18 x^{7} + 18 x^{5} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T509) $5832$ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 1,z^2 + 1$ $[1, 7, 25]$
3.1.18.40d1.10 $x^{18} + 6 x^{15} + 9 x^{5} + 3$ $\He_3^2:D_4$ (as 18T509) $5832$ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.11 $x^{18} + 6 x^{15} + 9 x^{8} + 9 x^{5} + 3$ $\He_3^2:D_4$ (as 18T509) $5832$ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.12 $x^{18} + 6 x^{15} + 18 x^{8} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.13 $x^{18} + 6 x^{15} + 9 x^{7} + 9 x^{5} + 3$ $\He_3^2:D_4$ (as 18T509) $5832$ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.14 $x^{18} + 6 x^{15} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.15 $x^{18} + 6 x^{15} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 3$ $\He_3^2:D_4$ (as 18T509) $5832$ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.16 $x^{18} + 6 x^{15} + 18 x^{7} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.17 $x^{18} + 6 x^{15} + 9 x^{8} + 18 x^{7} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.18 $x^{18} + 6 x^{15} + 18 x^{8} + 18 x^{7} + 9 x^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.19 $x^{18} + 6 x^{15} + 9 x^{5} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T509) $5832$ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.20 $x^{18} + 6 x^{15} + 9 x^{8} + 9 x^{5} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T509) $5832$ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.21 $x^{18} + 6 x^{15} + 18 x^{8} + 9 x^{5} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T509) $5832$ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.22 $x^{18} + 6 x^{15} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.23 $x^{18} + 6 x^{15} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.24 $x^{18} + 6 x^{15} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T509) $5832$ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.25 $x^{18} + 6 x^{15} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 3$ $\He_3^2:D_4$ (as 18T509) $5832$ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.26 $x^{18} + 6 x^{15} + 9 x^{8} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d1.27 $x^{18} + 6 x^{15} + 18 x^{8} + 18 x^{7} + 9 x^{5} + 9 x^{3} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 2]$ $z^9 + 2,2 z^3 + 2,2 z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.1 $x^{18} + 3 x^{15} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.2 $x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{5} + 3$ $C_3\wr D_4$ (as 18T189) $648$ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.3 $x^{18} + 3 x^{15} + 18 x^{9} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.4 $x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{5} + 3$ $S_3^2:S_3$ (as 18T103) $216$ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.5 $x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{8} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.6 $x^{18} + 3 x^{15} + 18 x^{9} + 9 x^{8} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.7 $x^{18} + 3 x^{15} + 18 x^{8} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.8 $x^{18} + 3 x^{15} + 9 x^{9} + 18 x^{8} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.9 $x^{18} + 3 x^{15} + 18 x^{9} + 18 x^{8} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.10 $x^{18} + 3 x^{15} + 9 x^{7} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.11 $x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{7} + 9 x^{5} + 3$ $C_3\wr D_4$ (as 18T189) $648$ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.12 $x^{18} + 3 x^{15} + 18 x^{9} + 9 x^{7} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.13 $x^{18} + 3 x^{15} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 3$ $C_3\wr D_4$ (as 18T189) $648$ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.14 $x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.15 $x^{18} + 3 x^{15} + 18 x^{9} + 9 x^{8} + 9 x^{7} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.16 $x^{18} + 3 x^{15} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 3$ $C_3\wr D_4$ (as 18T189) $648$ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.17 $x^{18} + 3 x^{15} + 9 x^{9} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 3$ $C_3\wr D_4$ (as 18T189) $648$ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.18 $x^{18} + 3 x^{15} + 18 x^{9} + 18 x^{8} + 9 x^{7} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.19 $x^{18} + 3 x^{15} + 18 x^{7} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.20 $x^{18} + 3 x^{15} + 9 x^{9} + 18 x^{7} + 9 x^{5} + 3$ $C_3\wr D_4$ (as 18T189) $648$ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.21 $x^{18} + 3 x^{15} + 18 x^{9} + 18 x^{7} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.22 $x^{18} + 3 x^{15} + 9 x^{8} + 18 x^{7} + 9 x^{5} + 3$ $C_3\wr D_4$ (as 18T189) $648$ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
3.1.18.40d2.23 $x^{18} + 3 x^{15} + 9 x^{9} + 9 x^{8} + 18 x^{7} + 9 x^{5} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[23, 15, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^3 + 1,z^2 + 2$ $[1, 7, 25]$
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