The BSD-predicted index formula for computed Heegner points

Let $E$ be the elliptic curve of conductor $N$, let $D$ be a negative fundamental discriminant satisfying the hypotheses above, and let $E_D$ be the quadratic twist of $E$ by $D$. Write the computed Heegner point as $P=lG+T$, where $G$ is a generator of the free part of $E(\mathbf Q)$, $T$ is a torsion point, and $l$ is an integer. Let $\Omega_{\rm re}$ and $\Omega_{\rm vol}$ denote the real and complex periods, respectively; let $c_p$ be the Tamagawa numbers, including the factor at infinity; let ${\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}$ denote the Tate–Shafarevich group; let $w(D)$ be the number of units in $\mathbf Q(\sqrt D)$; and let $\omega(n)$ be the number of distinct \prime factors of $n$. The BSD-predicted index formula. With these notations, the index $l$ satisfies

$$l^2={\Omega_{\rm re}\over 4\Omega_{\rm vol}} \biggl({\prod_{p|N\infty} c_p\cdot\#{\text{ {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n} \selectfont Sh}}}}\biggr) {\sqrt{|D|}\over \#E(\mathbf Q)_{\rm tors}^2}L(E_D,1)\cdot \biggl({w(D)\over 2}\biggr)^2 2^{\omega(\gcd(D,N))}.$$

This formula is obtained by combining the Gross–Zagier height formula with the Birch–Swinnerton-Dyer conjecture. The existence of the required quadratic twist is proven by Bump, Friedberg and Hoffstein, so the conjectural status of this displayed relation is resolved in the source's account.