↑

Properties

Label 2.2.2.6a1.4-1.4.16b
Base 2.2.2.6a1.4
Degree \(4\)
e \(4\)
f \(1\)
c \(16\)

Related objects

Downloads

Learn more

Defining polynomial

$x^{4} + b_{15} \pi^{4} x^{3} + \left(c_{18} \pi^{5} + b_{14} \pi^{4} + b_{10} \pi^{3}\right) x^{2} + \left(b_{17} \pi^{5} + a_{13} \pi^{4}\right) x + c_{16} \pi^{5} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.2.2.6a1.4
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $16$
Absolute Artin slopes: $[3,4,\frac{17}{4}]$
Swan slopes: $[4,\frac{9}{2}]$
Means: $\langle2,\frac{13}{4}\rangle$
Rams: $(4,5)$
Field count: $768$ (complete)
Ambiguity: $4$
Mass: $768$
Absolute Mass: $384$

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 8), $C_4^2:D_4$ (show 8), $C_2\wr D_4$ (show 8), $C_4^2:D_4$ (show 8), $C_2^4.Q_{16}$ (show 32), $C_2^6.D_4$ (show 32), $C_2^6.D_4$ (show 32), $C_2^5.\OD_{16}$ (show 32), $C_2^3.C_2\wr C_4$ (show 64), $C_2^6:C_8$ (show 32), $(C_2^2\times C_4^2):D_4$ (show 16), $(C_2^2\times C_4^2):Q_8$ (show 16), $C_2^5.D_8$ (show 32), $C_2^5.\SD_{16}$ (show 32), $C_2^6:D_4$ (show 16), $C_2^5.(C_2\times D_4)$ (show 32), $C_2^6:Q_8$ (show 16), $C_2^5.(C_2\times D_4)$ (show 32), $C_2^5.(C_2\times D_4)$ (show 64), $C_2^4.C_2\wr C_4$ (show 128), $C_2^7.(C_2\times D_4)$ (show 64), $C_2^5.C_2\wr C_4$ (show 64) (incomplete)
Hidden Artin slopes: $[2,2,\frac{7}{2}]$ (show 32), $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ (show 320), $[\frac{17}{4},2,2,\frac{7}{2},\frac{7}{2}]_{2}$ (show 64), $[2,2,3,\frac{7}{2},\frac{7}{2},\frac{17}{4}]^{2}$ (show 64), $[2,2,\frac{7}{2},\frac{7}{2}]^{2}$ (show 32), $[2,2,3,4,\frac{7}{2}]_{2}$ (show 32), $[2,3,\frac{7}{2},4]^{2}$ (show 64), $[2,2,\frac{7}{2},\frac{7}{2}]$ (show 32), $[2,2,3,\frac{7}{2},4]^{2}$ (show 32), not computed (show 64), $[2,2,3,\frac{7}{2}]^{2}$ (show 32) (incomplete)
Indices of inseparability: $[21,16,8,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,3,7,15]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
Sort order Select

Showing all 16

  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
2.2.8.56b2.2009 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2010 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 24\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2011 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{2} + 24 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2012 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 24\right) ( x^{2} + x + 1 )^{2} + 24 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2013 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2014 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 24\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2015 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{2} + 24 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2016 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 24\right) ( x^{2} + x + 1 )^{2} + 24 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2025 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2026 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 24\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2027 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{2} + 24 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2028 $( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 24\right) ( x^{2} + x + 1 )^{2} + 24 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2029 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2030 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 24\right) ( x^{2} + x + 1 )^{2} + 8 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2031 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{2} + 24 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
2.2.8.56b2.2032 $( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 24\right) ( x^{2} + x + 1 )^{2} + 24 ( x^{2} + x + 1 ) + 8 x + 2$ $C_2^6:D_4$ (as 16T956) $512$ $2$ $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]^{2}$ $[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]^{2}$ $[2,2,\frac{7}{2},\frac{7}{2},\frac{17}{4}]$ $[1,1,\frac{5}{2},\frac{5}{2},\frac{13}{4}]$ $[21, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + t$ $[1, 3, 7, 15]$
  displayed columns for results