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 ?
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$\begingroup$ What do you mean by a family of curves of that type? $\endgroup$– Will SawinCommented Nov 11, 2012 at 3:06
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$\begingroup$ Also, duplicated at stackexchange. $\endgroup$– MattCommented Nov 11, 2012 at 3:36
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3$\begingroup$ One should be able to perform a descent via 2-isogeny by following almost exactly the steps section X.6 of Silverman's Arithmetic of Elliptic Curves which deals with $y^2 = x^3 + px$. This should give an upper bound on the rank as a function of p modulo some power of 2, offhand I would guess 8, but my intuition is an artifact of working with the case of full 2-torsion. There may be some congruence classes of p for which this upper bound is 0, giving that the rank is exactly 0. I would be surprised if this did not happen. $\endgroup$– James WeigandtCommented Nov 11, 2012 at 5:06
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