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Properties

Label 2.1.4.11a
Base 2.1.1.0a1.1
Degree \(4\)
e \(4\)
f \(1\)
c \(11\)

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Defining polynomial

$x^{4} + 8 b_{11} x^{3} + 4 b_{6} x^{2} + 8 b_{9} x + 8 c_{8} + 16 c_{12} + 2$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}$
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $11$
Artin slopes: $[3,4]$
Swan slopes: $[2,3]$
Means: $\langle1,2\rangle$
Rams: $(2,4)$
Field count: $20$ (complete)
Ambiguity: $4$
Mass: $8$
Absolute Mass: $8$

Diagrams

Varying

Indices of inseparability: $[8,4,0]$
Associated inertia: $[1,1]$
Jump Set: $[1,3,7]$

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
2.1.4.11a1.1 8 $x^{4} + 2$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, 4]$ $[1,2,3]$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.2 8 $x^{4} + 18$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, 4]$ $[1,2,3]$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.3 4 $x^{4} + 8 x + 2$ $D_{4}$ (as 4T3) $8$ $2$ $[3, 4]^{2}$ $[2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.4 4 $x^{4} + 8 x^{3} + 8 x + 2$ $D_{4}$ (as 4T3) $8$ $2$ $[3, 4]^{2}$ $[2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.5 8 $x^{4} + 10$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, 4]$ $[1,2,3]$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.6 8 $x^{4} + 26$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, 4]$ $[1,2,3]$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.7 4 $x^{4} + 8 x + 10$ $D_{4}$ (as 4T3) $8$ $2$ $[3, 4]^{2}$ $[2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.8 4 $x^{4} + 8 x^{3} + 8 x + 10$ $D_{4}$ (as 4T3) $8$ $2$ $[3, 4]^{2}$ $[2,3]^{2}$ $[\ ]^{2}$ $[\ ]^{2}$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.9 8 $x^{4} + 4 x^{2} + 2$ $C_4$ (as 4T1) $4$ $4$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.10 8 $x^{4} + 4 x^{2} + 18$ $C_4$ (as 4T1) $4$ $4$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.11 8 $x^{4} + 8 x^{3} + 4 x^{2} + 2$ $C_4$ (as 4T1) $4$ $4$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.12 8 $x^{4} + 8 x^{3} + 4 x^{2} + 18$ $C_4$ (as 4T1) $4$ $4$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.13 8 $x^{4} + 4 x^{2} + 8 x + 2$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, 4]$ $[1,2,3]$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.14 8 $x^{4} + 4 x^{2} + 8 x + 18$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, 4]$ $[1,2,3]$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.15 8 $x^{4} + 4 x^{2} + 10$ $C_4$ (as 4T1) $4$ $4$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.16 8 $x^{4} + 4 x^{2} + 26$ $C_4$ (as 4T1) $4$ $4$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.17 8 $x^{4} + 8 x^{3} + 4 x^{2} + 10$ $C_4$ (as 4T1) $4$ $4$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.18 8 $x^{4} + 8 x^{3} + 4 x^{2} + 26$ $C_4$ (as 4T1) $4$ $4$ $[3, 4]$ $[2,3]$ $[\ ]$ $[\ ]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.19 8 $x^{4} + 4 x^{2} + 8 x + 10$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, 4]$ $[1,2,3]$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
2.1.4.11a1.20 8 $x^{4} + 4 x^{2} + 8 x + 26$ $D_{4}$ (as 4T3) $8$ $2$ $[2, 3, 4]$ $[1,2,3]$ $[2]$ $[1]$ $[8, 4, 0]$ $[1, 1]$ $z^2 + 1,z + 1$ $[1, 3, 7]$
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