Conjecture on quadratic twists of optimal elliptic curves

Let $E$ be an optimal elliptic curve over $\mathbb{Q}$ of conductor $N$, and let $-D$ be a negative fundamental discriminant such that $D$ is coprime to $N$. Write $E_{-D}$ for the twist of $E$ by $-D$, and let ${\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}(E_{-D})$ denote its Shafarevich–Tate group. For each \prime $p\mid N$, let $c_p(E_{-D})$ be the order of the arithmetic component group of $E_{-D}$ at $p$. If $L(E_{-D},1)\ne0$, so that $E_{-D}(\mathbb{Q})$ is finite, then

The quadratic-twist divisibility conjecture. The quantity $|E_{-D}(\mathbb{Q})|^2$ divides

$$|{\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}(E_{-D})|\prod_{p\mid N}c_p(E_{-D}),$$

up to a power of $2$.

This weakens the corresponding coprimality hypothesis in the cited conjecture from the source and is an implication of the paper's period formula. The assertion concerns the predicted arithmetic relation between the rational points, the Shafarevich–Tate group, and the local component groups of quadratic twists.