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Properties

Label 2.1.4.8a
Base 2.1.1.0a1.1
Degree \(4\)
e \(4\)
f \(1\)
c \(8\)

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Defining polynomial

$x^{4} + 4 b_{6} x^{2} + 4 a_{5} x + 2$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}$
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Artin slopes: $[\frac{8}{3},\frac{8}{3}]$
Swan slopes: $[\frac{5}{3},\frac{5}{3}]$
Means: $\langle\frac{5}{6},\frac{5}{4}\rangle$
Rams: $(\frac{5}{3},\frac{5}{3})$
Field count: $2$ (complete)
Ambiguity: $1$
Mass: $2$
Absolute Mass: $2$

Diagrams

Varying

Indices of inseparability: $[5,4,0]$
Associated inertia: $[1]$
Jump Set: $[1,3,7]$

Galois groups and Hidden Artin slopes

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
2.1.4.8a1.1 2 $x^{4} + 4 x + 2$ $S_4$ (as 4T5) $24$ $1$ $[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[5, 4, 0]$ $[1]$ $z + 1$ $[1, 3, 7]$
2.1.4.8a1.2 2 $x^{4} + 4 x^{2} + 4 x + 2$ $S_4$ (as 4T5) $24$ $1$ $[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$ $[\frac{5}{3},\frac{5}{3}]_{3}^{2}$ $[\ ]^{2}_{3}$ $[\ ]^{2}_{3}$ $[5, 4, 0]$ $[1]$ $z + 1$ $[1, 3, 7]$
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