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Properties

Label 2.1.4.8b1.2-1.4.25a
Base 2.1.4.8b1.2
Degree \(4\)
e \(4\)
f \(1\)
c \(25\)

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Defining polynomial

$x^{4} + \left(b_{35} \pi^{9} + b_{31} \pi^{8} + b_{27} \pi^{7} + b_{23} \pi^{6}\right) x^{3} + \left(c_{38} \pi^{10} + b_{10} \pi^{3} + a_{6} \pi^{2}\right) x^{2} + \left(b_{37} \pi^{10} + b_{33} \pi^{9} + b_{29} \pi^{8} + b_{25} \pi^{7}\right) x + c_{12} \pi^{4} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: 2.1.4.8b1.2
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $25$
Absolute Artin slopes: $[2,3,3,\frac{37}{8}]$
Swan slopes: $[3,\frac{19}{2}]$
Means: $\langle\frac{3}{2},\frac{11}{2}\rangle$
Rams: $(3,16)$
Field count: $256$ (complete)
Ambiguity: $4$
Mass: $512$
Absolute Mass: $128$

Diagrams

Varying

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

Galois group: $C_2^6.C_2^3.C_2^2$ (show 128), $(C_2^2\times D_4^2).D_4$ (show 128) (incomplete)
Hidden Artin slopes: not computed (show 240), $[\frac{7}{2},4,4,\frac{17}{4},\frac{17}{4},\frac{9}{2}]^{2}$ (show 16) (incomplete)
Indices of inseparability: $[42,26,24,8,0]$
Associated inertia: $[1,2,1]$
Jump Set: $[1,5,10,32,48]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing 1-50 of 256

Next   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
2.1.16.57d1.129 $x^{16} + 4 x^{10} + 2 x^{8} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.130 $x^{16} + 20 x^{10} + 2 x^{8} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.131 $x^{16} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.132 $x^{16} + 20 x^{10} + 16 x^{9} + 2 x^{8} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.133 $x^{16} + 4 x^{10} + 2 x^{8} + 16 x^{7} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.134 $x^{16} + 20 x^{10} + 2 x^{8} + 16 x^{7} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.135 $x^{16} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{7} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.136 $x^{16} + 20 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{7} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.137 $x^{16} + 4 x^{10} + 2 x^{8} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.138 $x^{16} + 20 x^{10} + 2 x^{8} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.139 $x^{16} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.140 $x^{16} + 4 x^{10} + 2 x^{8} + 16 x^{7} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.141 $x^{16} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{7} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.142 $x^{16} + 20 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{7} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.143 $x^{16} + 8 x^{15} + 4 x^{10} + 2 x^{8} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.144 $x^{16} + 8 x^{15} + 20 x^{10} + 2 x^{8} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.145 $x^{16} + 8 x^{15} + 4 x^{10} + 2 x^{8} + 16 x^{7} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.146 $x^{16} + 8 x^{15} + 20 x^{10} + 2 x^{8} + 16 x^{7} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.147 $x^{16} + 8 x^{15} + 4 x^{10} + 2 x^{8} + 16 x^{5} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.148 $x^{16} + 8 x^{15} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{5} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.149 $x^{16} + 8 x^{15} + 4 x^{10} + 2 x^{8} + 16 x^{7} + 16 x^{5} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.150 $x^{16} + 8 x^{15} + 20 x^{10} + 2 x^{8} + 16 x^{7} + 16 x^{5} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.151 $x^{16} + 8 x^{15} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{7} + 16 x^{5} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.152 $x^{16} + 8 x^{15} + 4 x^{10} + 2 x^{8} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.153 $x^{16} + 8 x^{15} + 20 x^{10} + 2 x^{8} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.154 $x^{16} + 8 x^{15} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.155 $x^{16} + 8 x^{15} + 20 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.156 $x^{16} + 8 x^{15} + 4 x^{10} + 2 x^{8} + 16 x^{7} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.157 $x^{16} + 8 x^{15} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{7} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.158 $x^{16} + 8 x^{13} + 4 x^{10} + 2 x^{8} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.159 $x^{16} + 8 x^{13} + 20 x^{10} + 2 x^{8} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.160 $x^{16} + 8 x^{13} + 4 x^{10} + 2 x^{8} + 16 x^{7} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.161 $x^{16} + 8 x^{13} + 20 x^{10} + 2 x^{8} + 16 x^{7} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.162 $x^{16} + 8 x^{13} + 4 x^{10} + 2 x^{8} + 16 x^{5} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.163 $x^{16} + 8 x^{13} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{5} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.164 $x^{16} + 8 x^{13} + 4 x^{10} + 2 x^{8} + 16 x^{7} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.165 $x^{16} + 8 x^{13} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{7} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.166 $x^{16} + 8 x^{15} + 8 x^{13} + 4 x^{10} + 2 x^{8} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.167 $x^{16} + 8 x^{15} + 8 x^{13} + 20 x^{10} + 2 x^{8} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.168 $x^{16} + 8 x^{15} + 8 x^{13} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.169 $x^{16} + 8 x^{15} + 8 x^{13} + 20 x^{10} + 16 x^{9} + 2 x^{8} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.170 $x^{16} + 8 x^{15} + 8 x^{13} + 4 x^{10} + 2 x^{8} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.171 $x^{16} + 8 x^{15} + 8 x^{13} + 20 x^{10} + 2 x^{8} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.172 $x^{16} + 8 x^{15} + 8 x^{13} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.173 $x^{16} + 8 x^{15} + 8 x^{13} + 20 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.174 $x^{16} + 8 x^{15} + 8 x^{13} + 4 x^{10} + 2 x^{8} + 16 x^{7} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.175 $x^{16} + 8 x^{15} + 8 x^{13} + 20 x^{10} + 2 x^{8} + 16 x^{7} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.176 $x^{16} + 8 x^{15} + 8 x^{13} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{7} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.177 $x^{16} + 8 x^{15} + 8 x^{13} + 20 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{7} + 16 x^{3} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
2.1.16.57d1.178 $x^{16} + 8 x^{11} + 4 x^{10} + 16 x^{9} + 2 x^{8} + 16 x^{7} + 16 x^{5} + 10$ $C_2^6.C_2^3.C_2^2$ (as 16T1372) $2048$ $2$ not computed not computed not computed not computed $[42, 26, 24, 8, 0]$ $[1, 2, 1]$ $z^8 + 1,z^6 + 1,z + 1$ $[1, 5, 10, 32, 48]$
Next   displayed columns for results