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Packages » EllipticCurves :: bruteForceRationalPoints(EllipticCurveW)
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bruteForceRationalPoints(EllipticCurveW) -- A brute-force method to list the points on an elliptic curve.

Synopsis

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

Ways to use this method: