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Properties

Label 2.2.4.16b2.8-1.2.4a
Base 2.2.4.16b2.8
Degree \(2\)
e \(2\)
f \(1\)
c \(4\)

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Defining polynomial

$x^{2} + \left(b_{5} \pi^{3} + a_{3} \pi^{2}\right) x + c_{6} \pi^{4} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.2.4.16b2.8
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $4$
Absolute Artin slopes: $[2,3,3]$
Swan slopes: $[3]$
Means: $\langle\frac{3}{2}\rangle$
Rams: $(3)$
Field count: $10$ (incomplete)
Ambiguity: $2$
Mass: $12$
Absolute Mass: $3$ ($47/16$ 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: $C_8: C_2$ (show 1), $C_2 \times (C_8:C_2)$ (show 1), $C_2^2:\OD_{16}$ (show 2), $C_2^3:\OD_{16}$ (show 2), $C_2^5:\OD_{16}$ (show 4) (incomplete)
Hidden Artin slopes: $[\ ]^{2}$ (show 1), $[2,2,3,3]^{2}$ (show 4), $[2,2]^{2}$ (show 2), $[2]^{2}$ (show 2), $[\ ]$ (show 1) (incomplete)
Indices of inseparability: $[13,10,4,0]$ (show 4), $[13,12,4,0]$ (show 6)
Associated inertia: $[1,1]$ (show 6), $[1,2]$ (show 4)
Jump Set: $[1,2,4,16]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 10

  displayed columns for results
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
2.2.8.40d1.29 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 4 x + 10$ $C_2^3:\OD_{16}$ (as 16T252) $128$ $4$ $[2, 2, 2, 3, 3]^{4}$ $[1,1,1,2,2]^{4}$ $[2,2]^{2}$ $[1,1]^{2}$ $[13, 12, 4, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 2, 4, 16]$
2.2.8.40d1.30 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 4 x + 2$ $C_2^2:\OD_{16}$ (as 16T95) $64$ $8$ $[2, 2, 3, 3]^{4}$ $[1,1,2,2]^{4}$ $[2]^{2}$ $[1]^{2}$ $[13, 12, 4, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 2, 4, 16]$
2.2.8.40d1.36 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 4 x + 10$ $C_2^3:\OD_{16}$ (as 16T252) $128$ $4$ $[2, 2, 2, 3, 3]^{4}$ $[1,1,1,2,2]^{4}$ $[2,2]^{2}$ $[1,1]^{2}$ $[13, 12, 4, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 2, 4, 16]$
2.2.8.40d1.39 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 4 x + 2$ $C_2^2:\OD_{16}$ (as 16T95) $64$ $8$ $[2, 2, 3, 3]^{4}$ $[1,1,2,2]^{4}$ $[2]^{2}$ $[1]^{2}$ $[13, 12, 4, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 2, 4, 16]$
2.2.8.40d1.43 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 4 x + 2$ $C_8: C_2$ (as 16T6) $16$ $16$ $[2, 3, 3]^{2}$ $[1,2,2]^{2}$ $[\ ]$ $[\ ]$ $[13, 12, 4, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 2, 4, 16]$
2.2.8.40d1.44 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 12 x + 2$ $C_2 \times (C_8:C_2)$ (as 16T15) $32$ $8$ $[2, 3, 3]^{4}$ $[1,2,2]^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $[13, 12, 4, 0]$ $[1, 1]$ $z^4 + 1,z^3 + 1$ $[1, 2, 4, 16]$
2.2.8.40d4.21 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 ) + 4 x + 2$ $C_2^5:\OD_{16}$ (as 16T826) $512$ $2$ $[2, 2, 2, 3, 3, 3, 3]^{4}$ $[1,1,1,2,2,2,2]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[13, 10, 4, 0]$ $[1, 2]$ $z^4 + 1,z^3 + z + t$ $[1, 2, 4, 16]$
2.2.8.40d4.26 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 ) + 4 x + 10$ $C_2^5:\OD_{16}$ (as 16T826) $512$ $2$ $[2, 2, 2, 3, 3, 3, 3]^{4}$ $[1,1,1,2,2,2,2]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[13, 10, 4, 0]$ $[1, 2]$ $z^4 + 1,z^3 + z + t$ $[1, 2, 4, 16]$
2.2.8.40d7.26 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 4 x + 2$ $C_2^5:\OD_{16}$ (as 16T826) $512$ $2$ $[2, 2, 2, 3, 3, 3, 3]^{4}$ $[1,1,1,2,2,2,2]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[13, 10, 4, 0]$ $[1, 2]$ $z^4 + 1,z^3 + t z + (t + 1)$ $[1, 2, 4, 16]$
2.2.8.40d7.31 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 4 x + 2$ $C_2^5:\OD_{16}$ (as 16T826) $512$ $2$ $[2, 2, 2, 3, 3, 3, 3]^{4}$ $[1,1,1,2,2,2,2]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[13, 10, 4, 0]$ $[1, 2]$ $z^4 + 1,z^3 + t z + (t + 1)$ $[1, 2, 4, 16]$
  displayed columns for results