We Know that from a conjecture by Goldfeld says that half of all elliptic curves have rank zero. Are there any known infinite families of elliptic curves in form $y^2=x^3+p^2x$ where p is prime with rank 0 ?
Elliptic curves with form$ y^2$=$x^3$+$p^2$$x$
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0See also [here](http://math.stackexchange.com/questions/1078757/how-could-we-show-that-the-abelian-group-has-text-rank-0/1078813#1078813). – 2014-12-30