# bruteForceRationalPoints(EllipticCurveW) -- A brute-force method to list the points on an elliptic curve.

## Description

 i1 : E=ellCurve({1,2,3,4,6},GF(29)) o1 = Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 .)) o1 : EllipticCurveW i2 : bruteForceRationalPoints(E) o2 = {Point ({0, 1, 0}( on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = ------------------------------------------------------------------------ X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ({2, 12, 1}( on the ------------------------------------------------------------------------ (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( ------------------------------------------------------------------------ over(GF 29 .))))), Point ({4, -12, 1}( on the (Elliptic Curve defined by ------------------------------------------------------------------------ Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ({4, ------------------------------------------------------------------------ 5, 1}( on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 ------------------------------------------------------------------------ + 4 X + 6( over(GF 29 .))))), Point ({8, -2, 1}( on the (Elliptic Curve ------------------------------------------------------------------------ defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), ------------------------------------------------------------------------ Point ({8, -9, 1}( on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = ------------------------------------------------------------------------ X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ({-13, -13, 1}( on the ------------------------------------------------------------------------ (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( ------------------------------------------------------------------------ over(GF 29 .))))), Point ({-13, -6, 1}( on the (Elliptic Curve defined ------------------------------------------------------------------------ by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ------------------------------------------------------------------------ ({-5, -11, 1}( on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 ------------------------------------------------------------------------ + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ({-5, 13, 1}( on the ------------------------------------------------------------------------ (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( ------------------------------------------------------------------------ over(GF 29 .))))), Point ({-11, -3, 1}( on the (Elliptic Curve defined ------------------------------------------------------------------------ by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ------------------------------------------------------------------------ ({-11, 11, 1}( on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 ------------------------------------------------------------------------ + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ({7, 1, 1}( on the (Elliptic ------------------------------------------------------------------------ Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 ------------------------------------------------------------------------ .))))), Point ({7, -11, 1}( on the (Elliptic Curve defined by Y^2 + 1 XY ------------------------------------------------------------------------ + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ({-1, 1, 1}( on ------------------------------------------------------------------------ the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( ------------------------------------------------------------------------ over(GF 29 .))))), Point ({-1, -3, 1}( on the (Elliptic Curve defined by ------------------------------------------------------------------------ Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ({-2, ------------------------------------------------------------------------ 7, 1}( on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 ------------------------------------------------------------------------ + 4 X + 6( over(GF 29 .))))), Point ({-2, -8, 1}( on the (Elliptic Curve ------------------------------------------------------------------------ defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), ------------------------------------------------------------------------ Point ({-4, 12, 1}( on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = ------------------------------------------------------------------------ X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ({-4, -11, 1}( on the ------------------------------------------------------------------------ (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( ------------------------------------------------------------------------ over(GF 29 .))))), Point ({-8, 1, 1}( on the (Elliptic Curve defined by ------------------------------------------------------------------------ Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ({-8, ------------------------------------------------------------------------ 4, 1}( on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 ------------------------------------------------------------------------ + 4 X + 6( over(GF 29 .))))), Point ({-6, 10, 1}( on the (Elliptic Curve ------------------------------------------------------------------------ defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), ------------------------------------------------------------------------ Point ({-6, -7, 1}( on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = ------------------------------------------------------------------------ X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ({10, -10, 1}( on the ------------------------------------------------------------------------ (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( ------------------------------------------------------------------------ over(GF 29 .))))), Point ({10, -3, 1}( on the (Elliptic Curve defined by ------------------------------------------------------------------------ Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ------------------------------------------------------------------------ ({-14, 8, 1}( on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + ------------------------------------------------------------------------ 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ({-14, 3, 1}( on the (Elliptic ------------------------------------------------------------------------ Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 ------------------------------------------------------------------------ .))))), Point ({0, 12, 1}( on the (Elliptic Curve defined by Y^2 + 1 XY ------------------------------------------------------------------------ + 3 Y = X^3 + 2 X^2 + 4 X + 6( over(GF 29 .))))), Point ({0, 14, 1}( on ------------------------------------------------------------------------ the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6( ------------------------------------------------------------------------ over(GF 29 .)))))} o2 : List