Frey Curve

Author

Eric W. Weisstein
June 5, 2002

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/FreyCurve.nb.

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

Initialization

<<Graphics`ImplicitPlot`

EllipticDiscriminant[eqn_,{x_,y_}]:=Module[
{
a1=Coefficient[eqn,x y],
a3=Coefficient[eqn,y],
a2=-Coefficient[eqn,x,2],
a4=-Coefficient[eqn,x],
a6=-eqn/.{x->0,y->0},
b2,b4,b6,b8
},
b2=a1^2+4a2;
b4=2a4+a1 a3;
b6=a3^2+4a6;
b8=a1^2a6+4a2 a6-a1 a3 a4+a2^2a3^2-a4^2;
-b2^2b8-8b4^3-27b6^2+9b2 b4 b6
]

©2005 Wolfram Research, Inc. except for portions noted otherwise

Plots

Frey[p_,{a_,b_}]:=y^2==x(x-a^p)(x+b^p)

ImplicitPlot[Frey[#,{1,2}],{x,-10,10},PlotRange->{{-10,10},{-10,10}},Ticks->None]&/@Range[4]

Plots

Discriminant

Simplify[
EllipticDiscriminant[
y^2-Expand[x(x-a^p)(x+b^p)],
{x,y}
]
]

j-Invariant

Simplify[PowerExpand[
With[{c=(a^p+b^p)^(1/p)},
(a^(2p)+b^(2p)+a^p b^p)^3-(c^(2p)-a^p b^p)^3
]
]]

((((a^p) (b^p)+(b^(2 p))+(a^(2 p))))^3)==((((c^(2 p))-(a^p) (b^p)))^3)