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Properties

Label 2.2.4.16b2.1-1.2.6a
Base 2.2.4.16b2.1
Degree \(2\)
e \(2\)
f \(1\)
c \(6\)

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Defining polynomial

$x^{2} + \left(b_{9} \pi^{5} + b_{7} \pi^{4} + a_{5} \pi^{3}\right) x + c_{10} \pi^{6} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.2.4.16b2.1
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Absolute Artin slopes: $[2,3,\frac{7}{2}]$
Swan slopes: $[5]$
Means: $\langle\frac{5}{2}\rangle$
Rams: $(5)$
Field count: $28$ (complete)
Ambiguity: $2$
Mass: $48$
Absolute Mass: $12$

Diagrams

Varying

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

Galois group: $C_4^2:C_2$ (show 1), $C_4^2:C_2$ (show 1), $C_4:D_4$ (show 1), $C_2^2.D_4$ (show 1), $C_2^4.D_4$ (show 4), $C_2^6.(C_2\times C_4)$ (show 4), $C_2^6.C_2^3.C_2^2$ (show 16) (incomplete)
Hidden Artin slopes: $[3]$ (show 4), $[2,2,3,3]^{2}$ (show 4), $[2,3]^{2}$ (show 4), not computed (show 16) (incomplete)
Indices of inseparability: $[15,10,4,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,5,13,21]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 28

  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
2.2.8.44d3.25 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 5, 13, 21]$
2.2.8.44d3.26 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{6} + 10 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 5, 13, 21]$
2.2.8.44d3.27 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 12 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 5, 13, 21]$
2.2.8.44d3.28 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{6} + 10 ( x^{2} + x + 1 )^{4} + 12 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 5, 13, 21]$
2.2.8.44d3.29 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 5, 13, 21]$
2.2.8.44d3.30 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{6} + 10 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 5, 13, 21]$
2.2.8.44d3.31 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 12 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 5, 13, 21]$
2.2.8.44d3.32 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{6} + 10 ( x^{2} + x + 1 )^{4} + 12 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + 1$ $[1, 5, 13, 21]$
2.2.8.44d4.41 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + t$ $[1, 5, 13, 21]$
2.2.8.44d4.42 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 10 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + t$ $[1, 5, 13, 21]$
2.2.8.44d4.43 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + t$ $[1, 5, 13, 21]$
2.2.8.44d4.44 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 10 ( x^{2} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + t$ $[1, 5, 13, 21]$
2.2.8.44d4.45 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + t$ $[1, 5, 13, 21]$
2.2.8.44d4.46 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 10 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + t$ $[1, 5, 13, 21]$
2.2.8.44d4.47 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + t$ $[1, 5, 13, 21]$
2.2.8.44d4.48 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{6} + 10 ( x^{2} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + t$ $[1, 5, 13, 21]$
2.2.8.44d5.73 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_4$ (as 16T274) $128$ $4$ $[2, 2, 3, 3, \frac{7}{2}]^{4}$ $[1,1,2,2,\frac{5}{2}]^{4}$ $[2,3]^{2}$ $[1,2]^{2}$ $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 5, 13, 21]$
2.2.8.44d5.74 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_4$ (as 16T274) $128$ $4$ $[2, 2, 3, 3, \frac{7}{2}]^{4}$ $[1,1,2,2,\frac{5}{2}]^{4}$ $[2,3]^{2}$ $[1,2]^{2}$ $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 5, 13, 21]$
2.2.8.44d5.75 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 2$ $C_4:D_4$ (as 16T34) $32$ $4$ $[2, 3, 3, \frac{7}{2}]^{2}$ $[1,2,2,\frac{5}{2}]^{2}$ $[3]$ $[2]$ $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 5, 13, 21]$
2.2.8.44d5.76 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 2$ $C_4^2:C_2$ (as 16T27) $32$ $4$ $[2, 3, 3, \frac{7}{2}]^{2}$ $[1,2,2,\frac{5}{2}]^{2}$ $[3]$ $[2]$ $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 5, 13, 21]$
2.2.8.44d5.77 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 12 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_4$ (as 16T274) $128$ $4$ $[2, 2, 3, 3, \frac{7}{2}]^{4}$ $[1,1,2,2,\frac{5}{2}]^{4}$ $[2,3]^{2}$ $[1,2]^{2}$ $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 5, 13, 21]$
2.2.8.44d5.78 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{4} + 12 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 2$ $C_2^4.D_4$ (as 16T274) $128$ $4$ $[2, 2, 3, 3, \frac{7}{2}]^{4}$ $[1,1,2,2,\frac{5}{2}]^{4}$ $[2,3]^{2}$ $[1,2]^{2}$ $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 5, 13, 21]$
2.2.8.44d5.79 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 2$ $C_4^2:C_2$ (as 16T30) $32$ $4$ $[2, 3, 3, \frac{7}{2}]^{2}$ $[1,2,2,\frac{5}{2}]^{2}$ $[3]$ $[2]$ $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 5, 13, 21]$
2.2.8.44d5.80 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + \left(8 x + 2\right) ( x^{2} + x + 1 )^{4} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 2$ $C_2^2.D_4$ (as 16T37) $32$ $4$ $[2, 3, 3, \frac{7}{2}]^{2}$ $[1,2,2,\frac{5}{2}]^{2}$ $[3]$ $[2]$ $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 5, 13, 21]$
2.2.8.44d5.81 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $C_2^6.(C_2\times C_4)$ (as 16T825) $512$ $2$ $[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$ $[1,1,1,2,2,2,\frac{5}{2}]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 5, 13, 21]$
2.2.8.44d5.82 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $C_2^6.(C_2\times C_4)$ (as 16T825) $512$ $2$ $[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$ $[1,1,1,2,2,2,\frac{5}{2}]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 5, 13, 21]$
2.2.8.44d5.83 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 12 ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $C_2^6.(C_2\times C_4)$ (as 16T825) $512$ $2$ $[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$ $[1,1,1,2,2,2,\frac{5}{2}]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 5, 13, 21]$
2.2.8.44d5.84 $( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{3} + 4 x ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 2$ $C_2^6.(C_2\times C_4)$ (as 16T825) $512$ $2$ $[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$ $[1,1,1,2,2,2,\frac{5}{2}]^{4}$ $[2,2,3,3]^{2}$ $[1,1,2,2]^{2}$ $[15, 10, 4, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 5, 13, 21]$
  displayed columns for results