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Properties

Label 2.2.2.4a2.2-1.4.13a
Base 2.2.2.4a2.2
Degree \(4\)
e \(4\)
f \(1\)
c \(13\)

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Defining polynomial

$x^{4} + \left(b_{15} \pi^{4} + b_{11} \pi^{3}\right) x^{3} + \left(c_{18} \pi^{5} + a_{2} \pi\right) x^{2} + \left(b_{17} \pi^{5} + b_{13} \pi^{4}\right) x + c_{4} \pi^{2} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.2.2.4a2.2
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $13$
Absolute Artin slopes: $[2,2,\frac{15}{4}]$
Swan slopes: $[1,\frac{9}{2}]$
Means: $\langle\frac{1}{2},\frac{5}{2}\rangle$
Rams: $(1,8)$
Field count: $712$ (incomplete)
Ambiguity: $4$
Mass: $768$
Absolute Mass: $384$ ($352$ 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: $C_4^2:D_4$ (show 4), $C_2\wr D_4$ (show 8), $C_4^2:D_4$ (show 8), $C_4^2:D_4$ (show 4), $C_2^4.Q_{16}$ (show 16), $C_2^6.D_4$ (show 16), $C_2^6.D_4$ (show 16), $C_2^5.C_2\wr C_4$ (show 128), $C_2^7.C_2\wr C_4$ (show 128), $C_2^7.C_2\wr C_4$ (show 128), $C_2^7.C_2\wr C_4$ (show 128), $C_2^7.C_2\wr C_4$ (show 128) (incomplete)
Hidden Artin slopes: $[2,2,3,\frac{7}{2},\frac{7}{2},\frac{7}{2},\frac{7}{2},\frac{15}{4}]^{2}$ (show 256), $[3,3,3,\frac{7}{2},\frac{7}{2},\frac{7}{2}]^{2}$ (show 32), not computed (show 352), $[3,\frac{7}{2},\frac{7}{2}]$ (show 24), $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ (show 16), $[2,3,\frac{7}{2},\frac{7}{2}]^{2}$ (show 16), $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]$ (show 16) (incomplete)
Indices of inseparability: $[14,6,4,0]$ (show 512), $[14,6,6,0]$ (show 200)
Associated inertia: $[1,1]$ (show 200), $[2,1]$ (show 512)
Jump Set: $[1,2,7,15]$ (show 256), $[1,3,6,16]$ (show 128), $[1,3,7,15]$ (show 256), $[1,3,11,19]$ (show 72)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 16

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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
2.2.8.42a1.7 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.8 $( x^{2} + x + 1 )^{8} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.11 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 8 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.12 $( x^{2} + x + 1 )^{8} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 8 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.21 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.22 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{6} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.25 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.26 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.37 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.38 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.49 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 8 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.50 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 8 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.53 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 8 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.54 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 8 x ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.71 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
2.2.8.42a1.72 $( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{3} + 2$ $C_2^6.D_4$ (as 16T834) $512$ $2$ $[2, 2, \frac{5}{2}, 3, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}]^{4}$ $[1,1,\frac{3}{2},2,\frac{5}{2},\frac{5}{2},\frac{11}{4}]^{4}$ $[\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]^{2}$ $[\frac{3}{2},2,\frac{5}{2},\frac{5}{2}]^{2}$ $[14, 6, 6, 0]$ $[1, 1]$ $z^6 + 1,z + 1$ $[1, 3, 11, 19]$
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