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Properties

Label 2.1.4.11a1.7-2.2.18a
Base 2.1.4.11a1.7
Degree \(4\)
e \(2\)
f \(2\)
c \(18\)

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Defining polynomial over unramified subextension

$x^{2} + \left(b_{15} \pi^{8} + b_{13} \pi^{7} + b_{11} \pi^{6} + b_{9} \pi^{5}\right) x + c_{16} \pi^{9} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.11a1.7
Ramification index $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $18$
Absolute Artin slopes: $[3,4,5]$
Swan slopes: $[8]$
Means: $\langle4\rangle$
Rams: $(8)$
Field count: $86$ (incomplete)
Ambiguity: $4$
Mass: $256$
Absolute Mass: $64$ ($32$ currently in the LMFDB)

Diagrams

Varying

The following invariants arise for fields within the LMFDB; since not all fields in this family are stored, it may be incomplete.

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group: $QD_{16}$ (show 4), $D_8:C_2$ (show 2), $D_8:C_2$ (show 2), $Q_{16}:C_2$ (show 2), $D_4:D_4$ (show 4), $C_4^2:C_2^2$ (show 4), $C_4^2:D_4$ (show 4), $C_2\wr D_4$ (show 8), $C_4^2:D_4$ (show 8), $C_4^2:D_4$ (show 4), $C_4^2:D_4$ (show 4), $C_2^6.D_4$ (show 8), $C_2\wr D_4$ (show 32) (incomplete)
Hidden Artin slopes: $[\ ]$ (show 4), $[2]$ (show 6), $[2,\frac{7}{2}]$ (show 8), $[2,\frac{7}{2},\frac{17}{4}]$ (show 24), $[2,2,\frac{7}{2}]$ (show 4), not computed (show 32), $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ (show 8) (incomplete)
Indices of inseparability: $[24,16,8,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,3,7,15]$

Fields

Galois group Galois Artin Indices Hidden content
Assoc. Inertia Jump Set
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Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
2.2.8.62a1.1611 $( x^{2} + x + 1 )^{8} + 16 x ( x^{2} + x + 1 )^{7} + 8 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $D_8:C_2$ (as 16T47) $32$ $8$ $[2, 3, 4, 5]^{2}$ $[1,2,3,4]^{2}$ $[2]$ $[1]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.1612 $( x^{2} + x + 1 )^{8} + 16 x ( x^{2} + x + 1 )^{7} + 8 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 32 x + 2$ $D_8:C_2$ (as 16T47) $32$ $8$ $[2, 3, 4, 5]^{2}$ $[1,2,3,4]^{2}$ $[2]$ $[1]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.1619 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $D_8:C_2$ (as 16T35) $32$ $8$ $[2, 3, 4, 5]^{2}$ $[1,2,3,4]^{2}$ $[2]$ $[1]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.1620 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 32 x + 2$ $Q_{16}:C_2$ (as 16T50) $32$ $8$ $[2, 3, 4, 5]^{2}$ $[1,2,3,4]^{2}$ $[2]$ $[1]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.1621 $( x^{2} + x + 1 )^{8} + 16 x ( x^{2} + x + 1 )^{7} + 8 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $D_8:C_2$ (as 16T35) $32$ $8$ $[2, 3, 4, 5]^{2}$ $[1,2,3,4]^{2}$ $[2]$ $[1]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.1622 $( x^{2} + x + 1 )^{8} + 16 x ( x^{2} + x + 1 )^{7} + 8 ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 32 x + 2$ $Q_{16}:C_2$ (as 16T50) $32$ $8$ $[2, 3, 4, 5]^{2}$ $[1,2,3,4]^{2}$ $[2]$ $[1]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
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