↑

Properties

Label 3.2.9.26a
Base 3.1.1.0a1.1
Degree \(18\)
e \(9\)
f \(2\)
c \(26\)

Related objects

Downloads

Learn more

Defining polynomial over unramified subextension

$x^{9} + 3a_{5} x^{5} + 3$

Invariants

Residue field characteristic: $3$
Degree: $18$
Base field: $\Q_{3}$
Ramification index $e$: $9$
Residue field degree $f$: $2$
Discriminant exponent $c$: $26$
Artin slopes: $[\frac{13}{8},\frac{13}{8}]$
Swan slopes: $[\frac{5}{8},\frac{5}{8}]$
Means: $\langle\frac{5}{12},\frac{5}{9}\rangle$
Rams: $(\frac{5}{8},\frac{5}{8})$
Field count: $5$ (complete)
Ambiguity: $2$
Mass: $8$
Absolute Mass: $4$

Diagrams

Varying

Indices of inseparability: $[5,5,0]$
Associated inertia: $[1]$
Jump Set: undefined

Fields

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

Showing all 5

  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
3.2.9.26a1.1 $( x^{2} + 2 x + 2 )^{9} + 3 ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[5, 5, 0]$ $[1]$ $z + (2 t + 2)$ undefined
3.2.9.26a2.1 $( x^{2} + 2 x + 2 )^{9} + 6 ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $2$ not computed not computed not computed not computed $[5, 5, 0]$ $[1]$ $z + (t + 1)$ undefined
3.2.9.26a3.1 $( x^{2} + 2 x + 2 )^{9} + 3 x ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[5, 5, 0]$ $[1]$ $z + (t + 2)$ undefined
3.2.9.26a4.1 $( x^{2} + 2 x + 2 )^{9} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[5, 5, 0]$ $[1]$ $z + 1$ undefined
3.2.9.26a5.1 $( x^{2} + 2 x + 2 )^{9} + 6 x ( x^{2} + 2 x + 2 )^{5} + 3$ not computed $ $not computed$ $ $1$ not computed not computed not computed not computed $[5, 5, 0]$ $[1]$ $z + (2 t + 1)$ undefined
  displayed columns for results