On Silverman s conjecture for a family of elliptic curves

Let E be an elliptic curve over Q with the given Weierstrass equation y²=x³+ax+b. If D is a squarefree integer, then let E(D) denote the D-quadratic twist of E that is given by E(D):y²=x³+aD²x+bD³. Let E(D)(Q) be the group of Q-rational points of E(D). It is conjectured by J. Silverman that there are infinitely many primes p for which E(p)(Q) has positive rank, and there are infinitely many primes q for which E(q)(Q) has rank 0. In this paper, assuming the parity conjecture, we show that for infinitely many primes p, the elliptic curve E(p)n:y²=x³−np2x has odd rank and for infinitely many primes p, E(p)n(Q) has even rank, where n is a positive integer that can be written as biquadrates sums in two different ways, i.e., n=u^4+v^4=r^4+s^4, where u,v,r,s are positive integers such that gcd(u,v)=gcd(r,s)=1. More precisely, we prove that: if n can be written in two different ways as biquartic sums and p is prime, then under the assumption of the parity conjecture E(p)n(Q) has odd rank (and so a positive rank) as long as n is odd and p≡5,7(mod8) or n is even and p≡1(mod4). In the end, we also compute the ranks of some specific values of n and p explicitly.