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

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$E_{p,q}(\mathbb{Q})$ and ${\dim }_{{\mathbb{F}}_2}\text{Sha}\left(E_{p,q}\mathrm{/}\text{bbQ}\right)[2]$ 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.