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Properties

Label 2.2.2.6a
Base 2.1.1.0a1.1
Degree \(4\)
e \(2\)
f \(2\)
c \(6\)

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Defining polynomial over unramified subextension

$x^{2} + 4 b_{3} x + 8 c_{4} + 2$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}$
Ramification index $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $6$
Artin slopes: $[3]$
Swan slopes: $[2]$
Means: $\langle1\rangle$
Rams: $(2)$
Field count: $6$ (complete)
Ambiguity: $4$
Mass: $4$
Absolute Mass: $2$

Diagrams

Varying

Indices of inseparability: $[2,0]$
Associated inertia: $[1]$
Jump Set: $[1,3]$

Galois groups and Hidden Artin slopes

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Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set Fields per packet
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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
2.2.2.6a1.1 2 $( x^{2} + x + 1 )^{2} + 2$ $C_2^2$ (as 4T2) $4$ $4$ $[3]^{2}$ $[2]^{2}$ $[\ ]$ $[\ ]$ $[2, 0]$ $[1]$ $z + 1$ $[1, 3]$
2.2.2.6a1.2 2 $( x^{2} + x + 1 )^{2} + 8 x + 2$ $C_4$ (as 4T1) $4$ $4$ $[3]^{2}$ $[2]^{2}$ $[\ ]$ $[\ ]$ $[2, 0]$ $[1]$ $z + 1$ $[1, 3]$
2.2.2.6a1.3 2 $( x^{2} + x + 1 )^{2} + 4 x ( x^{2} + x + 1 ) + 2$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3]^{2}$ $[1,2]^{2}$ $[2]$ $[1]$ $[2, 0]$ $[1]$ $z + 1$ $[1, 3]$
2.2.2.6a1.4 2 $( x^{2} + x + 1 )^{2} + 4 x ( x^{2} + x + 1 ) + 8 x + 2$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3]^{2}$ $[1,2]^{2}$ $[2]$ $[1]$ $[2, 0]$ $[1]$ $z + 1$ $[1, 3]$
2.2.2.6a1.5 2 $( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 2$ $C_2^2$ (as 4T2) $4$ $4$ $[3]^{2}$ $[2]^{2}$ $[\ ]$ $[\ ]$ $[2, 0]$ $[1]$ $z + 1$ $[1, 3]$
2.2.2.6a1.6 2 $( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 8 x + 2$ $C_4$ (as 4T1) $4$ $4$ $[3]^{2}$ $[2]^{2}$ $[\ ]$ $[\ ]$ $[2, 0]$ $[1]$ $z + 1$ $[1, 3]$
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