Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$. Let $p\ge11$ be a prime of good ordinary reduction for $E$ and assume that $p$ does not divide the degree of a minimal modular parametrization $\varphi_E:X_0(N)\to E$.

It's known by a work of Mazur that in this setting $p$ does not divide the Manin constant of $E$.

Is it also true that $p$ does not divide the product of the Tamagawa numbers $C=\prod_{\ell|N}c_\ell(E)$?

If the answer is affirmative, can anyone give me a reference? Conversely, does anyone know a counterexample? Are there conditions implying the non divisibility?

Sorry if the question is silly.