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Properties

Label 2.1.4.11a1.5-2.2.18a
Base 2.1.4.11a1.5
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: 2.1.4.11a1.5
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: $152$ (complete)
Ambiguity: $4$
Mass: $256$
Absolute Mass: $64$

Diagrams

Varying

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

Galois group: $D_8:C_2$ (show 2), $D_8:C_2$ (show 2), $D_8:C_2$ (show 2), $C_2\times \SD_{16}$ (show 2), $C_4^2:C_2^2$ (show 4), $C_4^2:C_2^2$ (show 8), $\OD_{16}:D_4$ (show 4), $C_4^2:D_4$ (show 4), $C_4^2:D_4$ (show 8), $C_4^2.D_4$ (show 8), $C_4^2:D_4$ (show 4), $C_2\wr D_4$ (show 8), $C_2^5.(C_2\times D_4)$ (show 16), $C_2^4.(C_4\times D_4)$ (show 16), $C_2^7.(C_2\times D_4)$ (show 64) (incomplete)
Hidden Artin slopes: $[2]$ (show 8), $[2,2,\frac{7}{2}]$ (show 4), $[2,\frac{7}{2},\frac{17}{4}]$ (show 24), $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4},\frac{17}{4},\frac{19}{4}]$ (show 8), $[2,3,\frac{7}{2},4,\frac{17}{4}]$ (show 16), not computed (show 56), $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ (show 16), $[2,\frac{7}{2}]$ (show 12), $[2,3,\frac{7}{2}]$ (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
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
2.2.8.62a1.269 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.270 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 ) + 48 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.271 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.272 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 ) + 48 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.273 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.274 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.275 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.276 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 ) + 48 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.277 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.278 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 ) + 48 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.299 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.300 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.301 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.302 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + 8 x ( x^{2} + x + 1 )^{6} + 16 ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 )^{3} + 16 ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.571 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.2.8.62a1.572 $( x^{2} + x + 1 )^{8} + 24 ( x^{2} + x + 1 )^{7} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{6} + 16 x ( x^{2} + x + 1 )^{5} + 16 ( x^{2} + x + 1 ) + 16 x + 2$ $C_2^4.(C_4\times D_4)$ (as 16T822) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, 5]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},4]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[24, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
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