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EllipticCurves :: bruteForceRationalPoints(EllipticCurve)

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

Synopsis

Description

i1 : E=ellCurve(3,8,GF(13))

o1 = Elliptic Curve defined by Y^2 = X^3 + 3 X + (-5)( over(GF 13 .))

o1 : EllipticCurve
i2 : bruteForceRationalPoints(E)

o2 = {Point ({0, 1, 0}( on the (Elliptic Curve defined by Y^2 = X^3 + 3 X +
     ------------------------------------------------------------------------
     (-5)( over(GF 13 .))))), Point ({1, -5, 1}( on the (Elliptic Curve
     ------------------------------------------------------------------------
     defined by Y^2 = X^3 + 3 X + (-5)( over(GF 13 .))))), Point ({1, 5, 1}(
     ------------------------------------------------------------------------
     on the (Elliptic Curve defined by Y^2 = X^3 + 3 X + (-5)( over(GF 13
     ------------------------------------------------------------------------
     .))))), Point ({2, 3, 1}( on the (Elliptic Curve defined by Y^2 = X^3 +
     ------------------------------------------------------------------------
     3 X + (-5)( over(GF 13 .))))), Point ({2, -3, 1}( on the (Elliptic Curve
     ------------------------------------------------------------------------
     defined by Y^2 = X^3 + 3 X + (-5)( over(GF 13 .))))), Point ({-1, 2, 1}(
     ------------------------------------------------------------------------
     on the (Elliptic Curve defined by Y^2 = X^3 + 3 X + (-5)( over(GF 13
     ------------------------------------------------------------------------
     .))))), Point ({-1, -2, 1}( on the (Elliptic Curve defined by Y^2 = X^3
     ------------------------------------------------------------------------
     + 3 X + (-5)( over(GF 13 .))))), Point ({-4, 6, 1}( on the (Elliptic
     ------------------------------------------------------------------------
     Curve defined by Y^2 = X^3 + 3 X + (-5)( over(GF 13 .))))), Point ({-4,
     ------------------------------------------------------------------------
     -6, 1}( on the (Elliptic Curve defined by Y^2 = X^3 + 3 X + (-5)(
     ------------------------------------------------------------------------
     over(GF 13 .)))))}

o2 : List