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Properties

Label 2.2.2.4a2.1-1.4.6a
Base 2.2.2.4a2.1
Degree \(4\)
e \(4\)
f \(1\)
c \(6\)

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Defining polynomial

$x^{4} + a_{3} \pi x^{3} + b_{2} \pi x^{2} + c_{4} \pi^{2} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.2.2.4a2.1
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Absolute Artin slopes: $[2,2,2]$
Swan slopes: $[1,1]$
Means: $\langle\frac{1}{2},\frac{3}{4}\rangle$
Rams: $(1,1)$
Field count: $14$ (incomplete)
Ambiguity: $4$
Mass: $12$
Absolute Mass: $6$ ($45/8$ 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: $C_2^3:C_4$ (show 1), $\OD_{16}:C_2$ (show 1), $C_2\wr C_4$ (show 1), $C_2\wr C_4$ (show 2), $D_4\times A_4$ (show 1), $C_2\wr C_6$ (show 4), $C_2^6:C_8$ (show 4) (incomplete)
Hidden Artin slopes: $[2,2,2]^{3}$ (show 4), $[2,2,2]^{4}$ (show 4), $[\ ]^{2}$ (show 2), $[2]^{2}$ (show 3), $[2]^{3}$ (show 1) (incomplete)
Indices of inseparability: $[7,6,4,0]$ (show 9), $[7,6,6,0]$ (show 2), $[7,7,4,0]$ (show 2), $[7,7,7,0]$ (show 1)
Associated inertia: $[2]$ (show 5), $[3]$ (show 5), $[4]$ (show 4)
Jump Set: $[1,2,4,15]$ (show 1), $[1,2,7,14]$ (show 1), $[1,2,7,15]$ (show 1), $[1,2,7,16]$ (show 1), $[1,3,6,15]$ (show 2), $[1,3,7,14]$ (show 1), $[1,3,7,15]$ (show 5), $[1,3,7,16]$ (show 2)

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 14

  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
2.2.8.28a2.1 $( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{7} + 2$ $C_2\wr C_6$ (as 16T719) $384$ $2$ $[2, 2, 2, 2, 2, 2]^{6}$ $[1,1,1,1,1,1]^{6}$ $[2,2,2]^{3}$ $[1,1,1]^{3}$ $[7, 7, 7, 0]$ $[3]$ $z^7 + t$ $[1, 3, 7, 15]$
2.2.8.28a5.1 $( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{6} + 2$ $C_2^6:C_8$ (as 16T924) $512$ $2$ $[2, 2, 2, 2, 2, 2]^{8}$ $[1,1,1,1,1,1]^{8}$ $[2,2,2]^{4}$ $[1,1,1]^{4}$ $[7, 6, 6, 0]$ $[4]$ $z^7 + t z + 1$ $[1, 3, 7, 15]$
2.2.8.28a7.2 $( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 6$ $C_2^6:C_8$ (as 16T924) $512$ $2$ $[2, 2, 2, 2, 2, 2]^{8}$ $[1,1,1,1,1,1]^{8}$ $[2,2,2]^{4}$ $[1,1,1]^{4}$ $[7, 6, 6, 0]$ $[4]$ $z^7 + t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.28a9.1 $( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{4} + 2$ $C_2^6:C_8$ (as 16T924) $512$ $2$ $[2, 2, 2, 2, 2, 2]^{8}$ $[1,1,1,1,1,1]^{8}$ $[2,2,2]^{4}$ $[1,1,1]^{4}$ $[7, 7, 4, 0]$ $[4]$ $z^7 + z^3 + t$ $[1, 2, 4, 15]$
2.2.8.28a10.2 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 x + 2$ $\OD_{16}:C_2$ (as 16T41) $32$ $4$ $[2, 2, 2]^{4}$ $[1,1,1]^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $[7, 6, 4, 0]$ $[2]$ $z^7 + z^3 + z + 1$ $[1, 2, 7, 16]$
2.2.8.28a10.3 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 6$ $C_2^3:C_4$ (as 16T33) $32$ $8$ $[2, 2, 2]^{4}$ $[1,1,1]^{4}$ $[\ ]^{2}$ $[\ ]^{2}$ $[7, 6, 4, 0]$ $[2]$ $z^7 + z^3 + z + 1$ $[1, 2, 7, 14]$
2.2.8.28a13.2 $( x^{2} + x + 1 )^{8} + \left(2 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 6$ $C_2\wr C_6$ (as 16T719) $384$ $2$ $[2, 2, 2, 2, 2, 2]^{6}$ $[1,1,1,1,1,1]^{6}$ $[2,2,2]^{3}$ $[1,1,1]^{3}$ $[7, 6, 4, 0]$ $[3]$ $z^7 + z^3 + t z + t$ $[1, 2, 7, 15]$
2.2.8.28a15.1 $( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 2$ $C_2^6:C_8$ (as 16T924) $512$ $2$ $[2, 2, 2, 2, 2, 2]^{8}$ $[1,1,1,1,1,1]^{8}$ $[2,2,2]^{4}$ $[1,1,1]^{4}$ $[7, 7, 4, 0]$ $[4]$ $z^7 + t z^3 + 1$ $[1, 3, 7, 15]$
2.2.8.28a19.2 $( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 4 x + 2$ $C_2\wr C_4$ (as 16T172) $64$ $4$ $[2, 2, 2, 2]^{4}$ $[1,1,1,1]^{4}$ $[2]^{2}$ $[1]^{2}$ $[7, 6, 4, 0]$ $[2]$ $z^7 + t z^3 + z + t$ $[1, 3, 7, 16]$
2.2.8.28a19.3 $( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 6$ $C_2\wr C_4$ (as 16T171) $64$ $4$ $[2, 2, 2, 2]^{4}$ $[1,1,1,1]^{4}$ $[2]^{2}$ $[1]^{2}$ $[7, 6, 4, 0]$ $[2]$ $z^7 + t z^3 + z + t$ $[1, 3, 7, 14]$
2.2.8.28a19.4 $( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 4 x + 6$ $C_2\wr C_4$ (as 16T172) $64$ $4$ $[2, 2, 2, 2]^{4}$ $[1,1,1,1]^{4}$ $[2]^{2}$ $[1]^{2}$ $[7, 6, 4, 0]$ $[2]$ $z^7 + t z^3 + z + t$ $[1, 3, 7, 16]$
2.2.8.28a21.2 $( x^{2} + x + 1 )^{8} + \left(2 x + 2\right) ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 6$ $C_2\wr C_6$ (as 16T719) $384$ $2$ $[2, 2, 2, 2, 2, 2]^{6}$ $[1,1,1,1,1,1]^{6}$ $[2,2,2]^{3}$ $[1,1,1]^{3}$ $[7, 6, 4, 0]$ $[3]$ $z^7 + t z^3 + t z + 1$ $[1, 3, 6, 15]$
2.2.8.28a23.1 $( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{7} + \left(2 x + 2\right) ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 2$ $C_2\wr C_6$ (as 16T719) $384$ $2$ $[2, 2, 2, 2, 2, 2]^{6}$ $[1,1,1,1,1,1]^{6}$ $[2,2,2]^{3}$ $[1,1,1]^{3}$ $[7, 6, 4, 0]$ $[3]$ $z^7 + t z^3 + t z + (t + 1)$ $[1, 3, 6, 15]$
2.2.8.28a26.1 $( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 2$ $D_4\times A_4$ (as 16T179) $96$ $2$ $[2, 2, 2, 2]^{6}$ $[1,1,1,1]^{6}$ $[2]^{3}$ $[1]^{3}$ $[7, 6, 4, 0]$ $[3]$ $z^7 + t z^3 + (t + 1) z + (t + 1)$ $[1, 3, 7, 15]$
  displayed columns for results