Mordell Curve

Author

Eric W. Weisstein
February 20, 2012

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/MordellCurve.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/MordellCurve.html.

©2012 Wolfram Research, Inc. except for portions noted otherwise

(y^2)==(x^3)+n

solns[n_]:=Module[{pts,x,y},
pts=Reduce[y^2==x^3+n&&Abs[x]<1000&&Abs[y]<1000,{x,y},Integers];
If[pts===False,{},{x,y}/.{ToRules[pts]}]
]

GraphicsGrid[Partition[ContourPlot[y^2==x^3+#,{x,-3,3.5},{y,-6.5,6.5},PlotLabel->y^2==x^3+#,ContourStyle->Red,Epilog->{PointSize[.04],Point[solns[#]]}]&/@Range[9],3,3,{1,1},{}],ImageSize->400]

Can't solve general equation

Reduce[y^2==x^3+1,{x,y},Integers]

So do an explicit search

Developer`SetSystemOptions["ReduceOptions"->{"DiscreteSolutionBound"->1000}];

Table[{n,Reduce[y^2==x^3+n&&x<1000&&0<=y<1000,{x,y},Integers]},{n,20}]//ColumnForm

Developer`SetSystemOptions["ReduceOptions"->{"DiscreteSolutionBound"->10000}];

Table[{n,{x,y}/.{ToRules@Reduce[y^2==x^3+n&&0<=y<100000,{x,y},Integers]
}},{n,10}]//ColumnForm

Values of n with no solutions

Position[Table[Reduce[y^2==x^3+n&&0<=y<1000,{x,y},Integers],{n,50}],False]//Flatten

Properties

Curvature

SortBy[FullSimplify[Curvature[y^2-(x^3+n),{x,y}],(x|y)∈Reals&&n∈Integers],ByteCount]

ByteCount/@%

Inflection Points

Reduce[y^2-(x^3+n)==6 x (3 n+(y^2))==0,{x,y},Reals]

Reduce[y^2-(x^3+n)==6 x (3 n+(y^2))==0&&n∈Integers,{x,y},Reals]

InflectionPoints[y^2-(x^3+n),{x,y}]

Singular Points

SingularPoints[y^2-(x^3+n),{x,y}]

Slope

Slope[y^2==x^3+n,{x,y}]//FullSimplify