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Properties

Label 2.1.2.3a1.2-1.4.18a
Base 2.1.2.3a1.2
Degree \(4\)
e \(4\)
f \(1\)
c \(18\)

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Defining polynomial

$x^{4} + \left(b_{19} \pi^{5} + a_{15} \pi^{4}\right) x^{3} + \left(c_{22} \pi^{6} + b_{14} \pi^{4} + b_{10} \pi^{3}\right) x^{2} + \left(b_{21} \pi^{6} + b_{17} \pi^{5}\right) x + c_{16} \pi^{5} + \pi$

Invariants

Residue field characteristic: $2$
Degree: $4$
Base field: $\Q_{2}(\sqrt{-2\cdot 5})$
Ramification index $e$: $4$
Residue field degree $f$: $1$
Discriminant exponent $c$: $18$
Absolute Artin slopes: $[3,4,\frac{19}{4}]$
Swan slopes: $[4,\frac{11}{2}]$
Means: $\langle2,\frac{15}{4}\rangle$
Rams: $(4,7)$
Field count: $32$ (complete)
Ambiguity: $4$
Mass: $32$
Absolute Mass: $16$

Diagrams

Varying

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

Galois group: $C_2 \wr C_2\wr C_2$
Hidden Artin slopes: $[2,\frac{7}{2},\frac{17}{4}]^{2}$
Indices of inseparability: $[23,16,8,0]$
Associated inertia: $[1,1,1]$
Jump Set: $[1,3,7,15]$

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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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
2.1.8.30a1.33 $x^{8} + 8 x^{7} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.34 $x^{8} + 8 x^{7} + 16 x^{6} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.35 $x^{8} + 8 x^{7} + 16 x^{5} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.36 $x^{8} + 8 x^{7} + 16 x^{6} + 16 x^{5} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.37 $x^{8} + 8 x^{7} + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.38 $x^{8} + 8 x^{7} + 16 x^{6} + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.39 $x^{8} + 8 x^{7} + 16 x^{5} + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.40 $x^{8} + 8 x^{7} + 16 x^{6} + 16 x^{5} + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.41 $x^{8} + 8 x^{7} + 8 x^{6} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.42 $x^{8} + 8 x^{7} + 24 x^{6} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.43 $x^{8} + 8 x^{7} + 8 x^{6} + 16 x^{5} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.44 $x^{8} + 8 x^{7} + 24 x^{6} + 16 x^{5} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.45 $x^{8} + 8 x^{7} + 8 x^{6} + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.46 $x^{8} + 8 x^{7} + 24 x^{6} + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.47 $x^{8} + 8 x^{7} + 8 x^{6} + 16 x^{5} + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.48 $x^{8} + 8 x^{7} + 24 x^{6} + 16 x^{5} + 26$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.49 $x^{8} + 8 x^{7} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.50 $x^{8} + 8 x^{7} + 16 x^{6} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.51 $x^{8} + 8 x^{7} + 16 x^{5} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.52 $x^{8} + 8 x^{7} + 16 x^{6} + 16 x^{5} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.53 $x^{8} + 8 x^{7} + 8 x^{2} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.54 $x^{8} + 8 x^{7} + 16 x^{6} + 8 x^{2} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.55 $x^{8} + 8 x^{7} + 16 x^{5} + 8 x^{2} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.56 $x^{8} + 8 x^{7} + 16 x^{6} + 16 x^{5} + 8 x^{2} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.57 $x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.58 $x^{8} + 8 x^{7} + 24 x^{6} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.59 $x^{8} + 8 x^{7} + 8 x^{6} + 16 x^{5} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.60 $x^{8} + 8 x^{7} + 24 x^{6} + 16 x^{5} + 8 x^{2} + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.61 $x^{8} + 8 x^{7} + 8 x^{6} + 8 x^{2} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.62 $x^{8} + 8 x^{7} + 24 x^{6} + 8 x^{2} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.63 $x^{8} + 8 x^{7} + 8 x^{6} + 16 x^{5} + 8 x^{2} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
2.1.8.30a1.64 $x^{8} + 8 x^{7} + 24 x^{6} + 16 x^{5} + 8 x^{2} + 16 x + 10$ $C_2 \wr C_2\wr C_2$ (as 8T35) $128$ $2$ $[2, 3, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]^{2}$ $[1,2,\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]^{2}$ $[2,\frac{7}{2},\frac{17}{4}]^{2}$ $[1,\frac{5}{2},\frac{13}{4}]^{2}$ $[23, 16, 8, 0]$ $[1, 1, 1]$ $z^4 + 1,z^2 + 1,z + 1$ $[1, 3, 7, 15]$
  displayed columns for results