↑

Properties

Label 5.2.4.6a
Base 5.1.1.0a1.1
Degree \(8\)
e \(4\)
f \(2\)
c \(6\)

Related objects

Downloads

Learn more

Defining polynomial over unramified subextension

$x^{4} + 5d_{0}$

Invariants

Residue field characteristic: $5$
Degree: $8$
Base field: $\Q_{5}$
Ramification index $e$: $4$
Residue field degree $f$: $2$
Discriminant exponent $c$: $6$
Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $4$ (complete)
Ambiguity: $8$
Mass: $1$
Absolute Mass: $1/2$

Varying

Indices of inseparability: $[0]$
Associated inertia: $[1]$
Jump Set: undefined (show 3), $[1]$ (show 1)

Galois groups and Hidden Artin slopes

Fields

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

Showing all 4

  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
5.2.4.6a1.1 2 $( x^{2} + 4 x + 2 )^{4} + 5 x$ $C_8$ (as 8T1) $8$ $8$ $[\ ]_{4}^{2}$ $[\ ]_{4}^{2}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^3 + 4 z^2 + z + 4$ undefined
5.2.4.6a1.2 2 $( x^{2} + 4 x + 2 )^{4} + 5$ $C_4\times C_2$ (as 8T2) $8$ $8$ $[\ ]_{4}^{2}$ $[\ ]_{4}^{2}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^3 + 4 z^2 + z + 4$ $[1]$
5.2.4.6a1.3 2 $( x^{2} + 4 x + 2 )^{4} + 5 x + 15$ $C_4\times C_2$ (as 8T2) $8$ $8$ $[\ ]_{4}^{2}$ $[\ ]_{4}^{2}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^3 + 4 z^2 + z + 4$ undefined
5.2.4.6a1.4 2 $( x^{2} + 4 x + 2 )^{4} + 20 x + 15$ $C_8$ (as 8T1) $8$ $8$ $[\ ]_{4}^{2}$ $[\ ]_{4}^{2}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^3 + 4 z^2 + z + 4$ undefined
  displayed columns for results