A note on a question of J. Nekovár and the Birch and Swinnerton-Dyer conjecture

If $D$ is a square-free integer, then let $E(D)$ denote the elliptic curve over $\mathbb{Q}$ given by the equation

Let $L(E(D),s)$ denote the Hasse-Weil $L$-function of $E(D)$, and let $L^{*}(E(D),1)$ denote the `algebraic part' of the central critical value $L(E(D),1)$. Using a theorem of Sturm, we verify a congruence conjectured by J. Neková\v{r}. By his work, if $S(3,E(D))$ denotes the 3-Selmer group of $E(D)$ and $D\neq 1$ is a square-free integer with $|D|\equiv 1\pmod 3$, then we find that