Proof of $L(E,1)/Ω(E)=1/8$ for elliptic curve $E:y^2=x^3-x/ \Bbb{Q}$?

Let

• $$E:y^2=x^3-x$$ be an elliptic curve over $$\Bbb{Q}$$ and
• $$ω_E=dx/2y=dx/2\sqrt{x^3-x}$$.

Then $$\begin{split} \Omega(E)&=\int_{E(\Bbb{R})} ω_E\\ \\ &=2\int\limits_1^{+\infty} dx/\sqrt{x^3-x} \end{split}$$ Question. Let the Hasse-Weil $$L$$-function be $$L(E,1)$$.
How can I prove that $$L(E,1)/\Omega(E)=1/8\text{ ?} \label{1}\tag{\star}$$ If you know any reference (★) is proved, I'll appreciate if you could show me how to calculate (★).

• You can do so by using module symbols, as it is the value of the modular symbol (suitably normalised with respect to your period) evaluated at 0. See this question here for how this is done. Cremona's book is a good source for the basics of how to verify BSD for elliptic curves. Commented Jul 16, 2022 at 11:29
• If we admit $L(E,1)＝0.65551438837・・・$ and $Ω(E)＝5.24411510858・・・$ , from this, can we say $L(E,1)/Ω(E)＝1/8$ without using computer ? Commented Jul 16, 2022 at 17:22
• In Cremona's book (somewhere around §2.8?), there is a proof using modular symbols that $L(E,1)/\Omega_E$ is a rational number with denominator bounded by $(p+1)-a_p$ for $p$ a prime of good reduction. Hence, approximating that number to this precision will give it you exactly.
– user471019
Commented Jul 16, 2022 at 18:23
• I'm confused. For $p＝2$, the elliptic curve has good reduction, so the denominator is smaller than $3$ ? Which prime $p$ you take ? Commented Jul 16, 2022 at 19:04
• It's equation (2.8.10) in Cremona. There is an additional factor of $2$ there. By the way, Magma tells me that the $L$-ratio is $1/4$. (It depends on how you define $\Omega_E$.)
– user471019
Commented Jul 16, 2022 at 21:53