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Properties

Label 2.2.4.20a2.19-1.2.8a
Base 2.2.4.20a2.19
Degree \(2\)
e \(2\)
f \(1\)
c \(8\)

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Defining polynomial

$x^{2} + \left(b_{13} \pi^{7} + b_{11} \pi^{6} + b_{9} \pi^{5} + a_{7} \pi^{4}\right) x + c_{14} \pi^{8} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $2$
Base field: 2.2.4.20a2.19
Ramification index $e$: $2$
Residue field degree $f$: $1$
Discriminant exponent $c$: $8$
Absolute Artin slopes: $[3,\frac{7}{2},\frac{9}{2}]$
Swan slopes: $[7]$
Means: $\langle\frac{7}{2}\rangle$
Rams: $(7)$
Field count: $192$ (complete)
Ambiguity: $2$
Mass: $192$
Absolute Mass: $96$

Diagrams

Varying

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

Galois group: $D_8:D_8$ (show 4), $D_8:D_8$ (show 4), $C_8^2:(C_2\times C_4)$ (show 8), $C_4^3.D_4$ (show 16), $\OD_{16}^2:(C_2\times C_4)$ (show 32), $C_2^7.D_4:D_4$ (show 64), $C_2^7.C_2\wr C_4$ (show 64) (incomplete)
Hidden Artin slopes: not computed (show 128), $[2,2,3,\frac{7}{2},\frac{9}{2}]$ (show 16), $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},4]^{2}$ (show 16), $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ (show 16), $[2,2,\frac{7}{2},\frac{9}{2}]$ (show 8), $[2,2,\frac{7}{2},\frac{9}{2}]^{2}$ (show 8) (incomplete)
Indices of inseparability: $[21,14,8,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,3,7,15]$

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
2.2.8.56a5.1161 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{2} + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1162 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{2} + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1163 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1164 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1165 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1166 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1167 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1168 $( x^{2} + x + 1 )^{8} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{4} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1177 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1178 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1179 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1180 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{4} + 16 x ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1181 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1182 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{4} + 16 ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1183 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
2.2.8.56a5.1184 $( x^{2} + x + 1 )^{8} + 12 ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{5} + \left(16 x + 4\right) ( x^{2} + x + 1 )^{4} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{3} + 8 ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 2$ $C_2^7.D_4:D_4$ (as 16T1713) $8192$ $2$ $[2, 2, 3, 3, 3, \frac{7}{2}, \frac{7}{2}, \frac{7}{2}, \frac{15}{4}, \frac{31}{8}, \frac{9}{2}]^{4}$ $[1,1,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8},\frac{7}{2}]^{4}$ $[2,2,3,3,\frac{7}{2},\frac{7}{2},\frac{15}{4},\frac{31}{8}]^{2}$ $[1,1,2,2,\frac{5}{2},\frac{5}{2},\frac{11}{4},\frac{23}{8}]^{2}$ $[21, 14, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + t,t z + (t + 1)$ $[1, 3, 7, 15]$
  displayed columns for results