↑

Properties

Label 3.1.18.46c
Base 3.1.1.0a1.1
Degree \(18\)
e \(18\)
f \(1\)
c \(46\)

Related objects

Downloads

Learn more

Defining polynomial

$x^{18} + 9 b_{34} x^{16} + 9 b_{32} x^{14} + 9 b_{31} x^{13} + 9 a_{29} x^{11} + 9 c_{27} x^{9} + 9 b_{24} x^{6} + 9 b_{21} x^{3} + 3 d_{0}$

Invariants

Residue field characteristic: $3$
Degree: $18$
Base field: $\Q_{3}$
Ramification index $e$: $18$
Residue field degree $f$: $1$
Discriminant exponent $c$: $46$
Artin slopes: $[\frac{5}{2},\frac{35}{12}]$
Swan slopes: $[\frac{3}{2},\frac{23}{12}]$
Means: $\langle1,\frac{29}{18}\rangle$
Rams: $(3,\frac{11}{2})$
Field count: $486$ (complete)
Ambiguity: $6$
Mass: $486$
Absolute Mass: $486$

Diagrams

Varying

Indices of inseparability: $[29,18,0]$
Associated inertia: $[1,1,1]$ (show 243), $[1,2,1]$ (show 243)
Jump Set: undefined (show 243), $[1,7,25]$ (show 243)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
Sort order Select

Showing 1-50 of 486

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
3.1.18.46c1.1 $x^{18} + 9 x^{11} + 3$ $C_3^3:\SOPlus(4,2)$ (as 18T351) $1944$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.2 $x^{18} + 9 x^{16} + 9 x^{11} + 3$ $C_3^3:\SOPlus(4,2)$ (as 18T351) $1944$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.3 $x^{18} + 18 x^{16} + 9 x^{11} + 3$ $C_3^3:\SOPlus(4,2)$ (as 18T351) $1944$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.4 $x^{18} + 9 x^{13} + 9 x^{11} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.5 $x^{18} + 9 x^{16} + 9 x^{13} + 9 x^{11} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.6 $x^{18} + 18 x^{16} + 9 x^{13} + 9 x^{11} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.7 $x^{18} + 18 x^{13} + 9 x^{11} + 3$ $C_3^3:\SOPlus(4,2)$ (as 18T351) $1944$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.8 $x^{18} + 9 x^{16} + 18 x^{13} + 9 x^{11} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.9 $x^{18} + 18 x^{16} + 18 x^{13} + 9 x^{11} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.10 $x^{18} + 9 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.11 $x^{18} + 9 x^{16} + 9 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.12 $x^{18} + 18 x^{16} + 9 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.13 $x^{18} + 9 x^{13} + 9 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.14 $x^{18} + 9 x^{16} + 9 x^{13} + 9 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.15 $x^{18} + 18 x^{16} + 9 x^{13} + 9 x^{11} + 9 x^{9} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.16 $x^{18} + 18 x^{13} + 9 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.17 $x^{18} + 9 x^{16} + 18 x^{13} + 9 x^{11} + 9 x^{9} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.18 $x^{18} + 18 x^{16} + 18 x^{13} + 9 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.19 $x^{18} + 18 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.20 $x^{18} + 9 x^{16} + 18 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.21 $x^{18} + 18 x^{16} + 18 x^{11} + 9 x^{9} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.22 $x^{18} + 9 x^{13} + 18 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.23 $x^{18} + 9 x^{16} + 9 x^{13} + 18 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.24 $x^{18} + 18 x^{16} + 9 x^{13} + 18 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.25 $x^{18} + 18 x^{13} + 18 x^{11} + 9 x^{9} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.26 $x^{18} + 9 x^{16} + 18 x^{13} + 18 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.27 $x^{18} + 18 x^{16} + 18 x^{13} + 18 x^{11} + 9 x^{9} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.28 $x^{18} + 9 x^{11} + 9 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.29 $x^{18} + 9 x^{16} + 9 x^{11} + 9 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.30 $x^{18} + 18 x^{16} + 9 x^{11} + 9 x^{6} + 3$ $C_3^3:\SOPlus(4,2)$ (as 18T351) $1944$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.31 $x^{18} + 9 x^{13} + 9 x^{11} + 9 x^{6} + 3$ $C_3^3:\SOPlus(4,2)$ (as 18T351) $1944$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.32 $x^{18} + 9 x^{16} + 9 x^{13} + 9 x^{11} + 9 x^{6} + 3$ $C_3^3:\SOPlus(4,2)$ (as 18T351) $1944$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.33 $x^{18} + 18 x^{16} + 9 x^{13} + 9 x^{11} + 9 x^{6} + 3$ $C_3^3:\SOPlus(4,2)$ (as 18T351) $1944$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.34 $x^{18} + 18 x^{13} + 9 x^{11} + 9 x^{6} + 3$ $C_3^3:\SOPlus(4,2)$ (as 18T351) $1944$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.35 $x^{18} + 9 x^{16} + 18 x^{13} + 9 x^{11} + 9 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.36 $x^{18} + 18 x^{16} + 18 x^{13} + 9 x^{11} + 9 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.37 $x^{18} + 9 x^{11} + 9 x^{9} + 9 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.38 $x^{18} + 9 x^{16} + 9 x^{11} + 9 x^{9} + 9 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.39 $x^{18} + 18 x^{16} + 9 x^{11} + 9 x^{9} + 9 x^{6} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.40 $x^{18} + 9 x^{13} + 9 x^{11} + 9 x^{9} + 9 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.41 $x^{18} + 9 x^{16} + 9 x^{13} + 9 x^{11} + 9 x^{9} + 9 x^{6} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.42 $x^{18} + 18 x^{16} + 9 x^{13} + 9 x^{11} + 9 x^{9} + 9 x^{6} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.43 $x^{18} + 18 x^{13} + 9 x^{11} + 9 x^{9} + 9 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.44 $x^{18} + 9 x^{16} + 18 x^{13} + 9 x^{11} + 9 x^{9} + 9 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.45 $x^{18} + 18 x^{16} + 18 x^{13} + 9 x^{11} + 9 x^{9} + 9 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 2$ $[1, 7, 25]$
3.1.18.46c1.46 $x^{18} + 18 x^{11} + 9 x^{9} + 9 x^{6} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.47 $x^{18} + 9 x^{16} + 18 x^{11} + 9 x^{9} + 9 x^{6} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.48 $x^{18} + 18 x^{16} + 18 x^{11} + 9 x^{9} + 9 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.49 $x^{18} + 9 x^{13} + 18 x^{11} + 9 x^{9} + 9 x^{6} + 3$ $\He_3^2:D_4$ (as 18T492) $5832$ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
3.1.18.46c1.50 $x^{18} + 9 x^{16} + 9 x^{13} + 18 x^{11} + 9 x^{9} + 9 x^{6} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[29, 18, 0]$ $[1, 1, 1]$ $z^9 + 2,2 z^6 + 1,z + 1$ $[1, 7, 25]$
Next   displayed columns for results