Let $E$ be an elliptic curve and $E_n$ be its quadratic twist by $n$. Let $\phi_n: X_0(N_n) \to E_n$ be the modular parametrization of $E_n$.

For some particular curves $E$, it seems (based on computing some examples in sage) to always be the case that $\phi_n(0) = O$ when the rank of $E_n$ is 0. For the curves I'm looking at, the torsion subgroup is $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ for all of the $E_n$. What are some possible explanations for this or strategies to go about proving it?

Let $E$ be an elliptic curve and $E_n$ be its quadratic twist by $n$. Let $\phi_n: X_0(N_n) \to E_n$ be the normalized modular parametrization of $E_n$ ($\infty \to O$)

For some particular curves $E$, it seems (based on computing some examples in sage) to always be the case that $\phi_n(0) = O$ when the rank of $E_n$ is 0. For the curves I'm looking at, the torsion subgroup is $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ for all of the $E_n$. What are some possible explanations for this or strategies to go about proving it?

Note: Edited to clarify the first comment.