↑

Properties

Label 3.3.6.27a
Base 3.1.1.0a1.1
Degree \(18\)
e \(6\)
f \(3\)
c \(27\)

Related objects

Downloads

Learn more

Defining polynomial over unramified subextension

$x^{6} + 3 b_{5} x^{5} + 3 a_{4} x^{4} + 3 d_{0} + 9 c_{6}$

Invariants

Residue field characteristic: $3$
Degree: $18$
Base field: $\Q_{3}$
Ramification index $e$: $6$
Residue field degree $f$: $3$
Discriminant exponent $c$: $27$
Artin slopes: $[2]$
Swan slopes: $[1]$
Means: $\langle\frac{2}{3}\rangle$
Rams: $(2)$
Field count: $496$ (complete)
Ambiguity: $18$
Mass: $702$
Absolute Mass: $234$

Diagrams

Varying

Indices of inseparability: $[4,0]$
Associated inertia: $[1,1]$ (show 372), $[1,2]$ (show 124)
Jump Set: undefined (show 248), $[1,3]$ (show 1), $[1,7]$ (show 230), $[1,8]$ (show 15), $[1,9]$ (show 2)

Fields

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

Showing all 15

  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
3.3.6.27a1.46 $( x^{3} + 2 x + 1 )^{6} + 3 x^{2} ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.47 $( x^{3} + 2 x + 1 )^{6} + 3 x^{2} ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 9 x^{2} + 3$ $C_3^3:C_{18}$ (as 18T159) $486$ $3$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.48 $( x^{3} + 2 x + 1 )^{6} + 3 x^{2} ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 18 x^{2} + 3$ $C_3^3:C_{18}$ (as 18T159) $486$ $3$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.49 $( x^{3} + 2 x + 1 )^{6} + 3 x ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.50 $( x^{3} + 2 x + 1 )^{6} + 3 x ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 9 x^{2} + 3$ $C_3^2:C_{18}$ (as 18T82) $162$ $3$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.51 $( x^{3} + 2 x + 1 )^{6} + 3 x ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 18 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.52 $( x^{3} + 2 x + 1 )^{6} + 3 ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 3$ $C_3^2\times S_3$ (as 18T17) $54$ $9$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.53 $( x^{3} + 2 x + 1 )^{6} + 3 ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 9 x^{2} + 3$ not computed $ $not computed$ $ $9$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.54 $( x^{3} + 2 x + 1 )^{6} + 3 ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 18 x^{2} + 3$ not computed $ $not computed$ $ $9$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.55 $( x^{3} + 2 x + 1 )^{6} + \left(3 x^{2} + 3\right) ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 3$ $C_3^4:C_6$ (as 18T162) $486$ $3$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.56 $( x^{3} + 2 x + 1 )^{6} + \left(3 x^{2} + 3\right) ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 9 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.57 $( x^{3} + 2 x + 1 )^{6} + \left(3 x^{2} + 3\right) ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 18 x^{2} + 3$ $C_3^3:C_{18}$ (as 18T159) $486$ $3$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.58 $( x^{3} + 2 x + 1 )^{6} + \left(3 x^{2} + 6\right) ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.59 $( x^{3} + 2 x + 1 )^{6} + \left(3 x^{2} + 6\right) ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 9 x^{2} + 3$ $C_3^3:C_{18}$ (as 18T159) $486$ $3$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
3.3.6.27a1.60 $( x^{3} + 2 x + 1 )^{6} + \left(3 x^{2} + 6\right) ( x^{3} + 2 x + 1 )^{5} + 6 ( x^{3} + 2 x + 1 )^{4} + 18 x^{2} + 3$ not computed $ $not computed$ $ $3$ not computed not computed not computed not computed $[4, 0]$ $[1, 1]$ $z^3 + 2,2 z^2 + 1$ $[1, 8]$
  displayed columns for results