In John Cremona's book, he defines the modular symbol of an elliptic curve in the following way.

Let $E/\mathbf{Q}$ be an elliptic curve and let $f_E$ be the modular form associated to $E$. The modular symbol associated to $E$ is the map $\mathbf{Q} \to \mathbf{Q}$ given by sending any $r \in \mathbf{Q}$ to the rational number

$$ [r] := \dfrac{2 \pi i}{\Omega_E} \left( \int_r^{i \infty} f_E(z) \, dz + \int_{-r}^{i \infty} f_E(z) \, dz \right). $$

These modular symbols contain interesting information about $L$-values of $E$. I have two questions about this (in increasing levels of importance):

1. Where can I find a proof that $[r]$ is indeed a rational number? If anyone has a reference, that would be great.

2. How do we compute the value of $[r]$ exactly? Cremona's book says that for any $r \in \mathbf{Q}$, the value of $[r]$ can be computed exactly. (And in any case, Sage can compute the values exactly.) But I can't find a theorem anywhere that gives a formula for the exact value of $[r]$. If anyone knows of an algorithm to get an exact value of $[r]$, or better yet, a formula that gives the exact value of $[r]$ in terms of invariants of the elliptic curve $E$, I'd greatly appreciate it.

Thanks for the help!

In John Cremona's book, he defines the modular symbol of an elliptic curve in the following way.

Let $E/\mathbf{Q}$ be an elliptic curve and let $f_E$ be the modular form associated to $E$. The modular symbol associated to $E$ is the map $\mathbf{Q} \to \mathbf{Q}$ given by sending any $r \in \mathbf{Q}$ to the rational number

$$ [r] := \dfrac{2 \pi i}{\Omega_E} \left( \int_r^{i \infty} f_E(z) \, dz + \int_{-r}^{i \infty} f_E(z) \, dz \right). $$

These modular symbols contain interesting information about $L$-values of $E$. I have two questions about this (in increasing levels of importance):

1. Where can I find a proof that $[r]$ is indeed a rational number? If anyone has a reference, that would be great.

2. How do we compute the value of $[r]$ exactly (i.e: not a numerical approximation)? Cremona's book says that for any $r \in \mathbf{Q}$, the value of $[r]$ can be computed exactly. (And in any case, Sage can compute the values exactly.) But I can't find a theorem anywhere that gives a formula for the exact value of $[r]$. If anyone knows of an algorithm to get an exact value of $[r]$, or better yet, a formula that gives the exact value of $[r]$ in terms of invariants of the elliptic curve $E$, I'd greatly appreciate it.

Thanks for the help!