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Properties

Label 2.2.4.16b1.4-1.2.9a
Base 2.2.4.16b1.4
Degree \(2\)
e \(2\)
f \(1\)
c \(9\)

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Defining polynomial

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

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.2.4.16b1.4
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $9$
Absolute Artin slopes: $[2,3,\frac{17}{4}]$
Swan slopes: $[8]$
Means: $\langle4\rangle$
Rams: $(8)$
Field count: $72$ (complete)
Ambiguity: $2$
Mass: $256$
Absolute Mass: $32$

Diagrams

Varying

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), $C_2^4.D_8$ (show 16), $C_2^6.D_4$ (show 16), $C_2^6.D_4$ (show 16)
Hidden Artin slopes: $[2,\frac{7}{2},\frac{7}{2},4]^{2}$ (show 16), $[2,\frac{7}{2},\frac{7}{2},4]$ (show 16), $[2,3,\frac{7}{2},4]^{2}$ (show 16), $[2,\frac{7}{2},4]$ (show 24)
Indices of inseparability: $[18,10,4,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,7,15,23]$

Fields

Galois group Galois Artin Indices Hidden content
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
2.2.8.50b1.302 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.303 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.307 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.308 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.313 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.314 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.317 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.318 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.319 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.320 $( x^{2} + x + 1 )^{8} + 8 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.321 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.322 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.323 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.324 $( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.326 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
2.2.8.50b1.327 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 8 x ( x^{2} + x + 1 )^{3} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_8$ (as 16T703) $256$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4}]^{2}$ $[2,\frac{7}{2},\frac{7}{2},4]$ $[1,\frac{5}{2},\frac{5}{2},3]$ $[18, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 7, 15, 23]$
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