Exact classification of elliptic curves $y^{2}=x^{3}-pqx$ with rank $0$ and trivial $\Sha[2]$

For the elliptic curves $E_{p,q}: y^{2}=x^{3}-pqx$ where $p$ and $q$ are distinct odd primes, we establish necessary and sufficient conditions under which $\rank E_{p,q}(\bbQ)$ and $\dim_{\mathbb{F}_{2}} \Sha \left( E_{p,q}/\bbQ \right)[2]$ are both $0$.

We do so via a similar characterisation of when the Selmer groups associated with the degree-$2$ isogeny $\phi$ and its dual $\widehat{\phi}$ are both of minimal size, along with results about a cokernel that arises from a related exact sequence.