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Properties

Label 2.1.4.9a
Base 2.1.1.0a1.1
Degree \(4\)
e \(4\)
f \(1\)
c \(9\)

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Defining polynomial

$x^{4} + 4 b_{7} x^{3} + \left(2 a_{2} + 8 c_{10}\right) x^{2} + 8 b_{9} x + 4 c_{4} + 2$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}$
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $9$
Artin slopes: $[2,\frac{7}{2}]$
Swan slopes: $[1,\frac{5}{2}]$
Means: $\langle\frac{1}{2},\frac{3}{2}\rangle$
Rams: $(1,4)$
Field count: $8$ (complete)
Ambiguity: $4$
Mass: $4$
Absolute Mass: $4$

Diagrams

Varying

Indices of inseparability: $[6,2,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,2,8]$ (show 4), $[1,5,9]$ (show 4)

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 8

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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
2.1.4.9a1.1 8 $x^{4} + 2 x^{2} + 2$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, \frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.1.4.9a1.2 8 $x^{4} + 10 x^{2} + 2$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, \frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.1.4.9a1.3 8 $x^{4} + 4 x^{3} + 2 x^{2} + 2$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, \frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.1.4.9a1.4 8 $x^{4} + 4 x^{3} + 10 x^{2} + 2$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, \frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 5, 9]$
2.1.4.9a1.5 8 $x^{4} + 2 x^{2} + 6$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, \frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 8]$
2.1.4.9a1.6 8 $x^{4} + 10 x^{2} + 6$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, \frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 8]$
2.1.4.9a1.7 8 $x^{4} + 2 x^{2} + 8 x + 6$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, \frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 8]$
2.1.4.9a1.8 8 $x^{4} + 4 x^{3} + 2 x^{2} + 8 x + 6$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, \frac{7}{2}]$ $[1,2,\frac{5}{2}]$ $[3]$ $[2]$ $[6, 2, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 2, 8]$
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