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Properties

Label 5.3.4.9a
Base 5.1.1.0a1.1
Degree \(12\)
e \(4\)
f \(3\)
c \(9\)

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Defining polynomial over unramified subextension

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

Invariants

Residue field characteristic: $5$
Degree: $12$
Base field: $\Q_{5}$
Ramification index $e$: $4$
Residue field degree $f$: $3$
Discriminant exponent $c$: $9$
Artin slopes: $[\ ]$
Swan slopes: $[\ ]$
Means: $\langle\ \rangle$
Rams: $(\ )$
Field count: $4$ (complete)
Ambiguity: $12$
Mass: $1$
Absolute Mass: $1/3$

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
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Showing all 4

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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
5.3.4.9a1.1 4 $( x^{3} + 3 x + 3 )^{4} + 5 x^{2}$ $C_{12}$ (as 12T1) $12$ $12$ $[\ ]_{4}^{3}$ $[\ ]_{4}^{3}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^3 + 4 z^2 + z + 4$ undefined
5.3.4.9a1.2 4 $( x^{3} + 3 x + 3 )^{4} + 5 x$ $C_{12}$ (as 12T1) $12$ $12$ $[\ ]_{4}^{3}$ $[\ ]_{4}^{3}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^3 + 4 z^2 + z + 4$ undefined
5.3.4.9a1.3 4 $( x^{3} + 3 x + 3 )^{4} + 5$ $C_{12}$ (as 12T1) $12$ $12$ $[\ ]_{4}^{3}$ $[\ ]_{4}^{3}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^3 + 4 z^2 + z + 4$ $[1]$
5.3.4.9a1.4 4 $( x^{3} + 3 x + 3 )^{4} + 10 x + 10$ $C_{12}$ (as 12T1) $12$ $12$ $[\ ]_{4}^{3}$ $[\ ]_{4}^{3}$ $[\ ]$ $[\ ]$ $[0]$ $[1]$ $z^3 + 4 z^2 + z + 4$ undefined
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