A finiteness criterion for the number of rational points for twisted elliptic Weil curves

We consider the Weil elliptic curve $E/\mathbb{Q}$ and let $L(E,s)=\sum^\infty_{n=1}a(n)n^{-s}$ be its canonical $L$-series. Admitting the Birch–Swinnerton–Dyer conjecture and fixing the curve $E$, a criterion is given for the finiteness of the group $E_D(\mathbb{Q})$ for twisted elliptic curves $E_D$, defined by the condition
$$ L(E_D,s)=\sum^\infty_{n=1}\chi(n)a(n)n^{-s}, $$
where $D$ is the discriminant of the quadratic field and $\chi(D)$ is its quadratic character.