on silverman's conjecture for a family of elliptic curves

let $e$ be an elliptic curve over $bbb{q}$ with the given weierstrass equation $ y^2=x^3+ax+b$. if $d$ is a squarefree integer, then let $e^{(d)}$ denote the $d$-quadratic twist of $e$ that is given by $e^{(d)}: y^2=x^3+ad^2x+bd^3$. let $e^{(d)}(bbb{q})$ be the group of $bbb{q}$-rational points of $e^{(d)}$. it is conjectured by j. silverman that there are infinitely many primes $p$ for which $e^{(p)}(bbb{q})$ has positive rank, and there are infinitely many primes $q$ for which $e^{(q)}(bbb{q})$ has rank $0$. in this paper, assuming the parity conjecture, we show that for infinitely many primes $p$, the elliptic curve $e_n^{(p)}: y^2=x^3-np^2x$ has odd rank and for infinitely many primes $p$, $e_n^{(p)}(bbb{q})$ 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 $e_n^{(p)}(bbb{q})$ has odd rank (and so a positive rank) as long as $n$ is odd and $pequiv5, 7pmod{8}$ or $n$ is even and $pequiv1pmod{4}$. in the end, we also compute the ranks of some specific values of $n$ and $p$ explicitly.