p-adic Valuation of L-Values for Elliptic Curves

Question

I'm wondering if the following conjecture is true:

Let $\mathcal{A}$ be an isogeny class of elliptic curves over $\mathbf{Q}$. Fix an odd prime $p$ of good reduction. Then there is a curve $E \in \mathcal{A}$ such that the quantity $$\dfrac{L(E,1)}{\Omega_E}$$ is a $p$-adic unit.

That is, the quantity $\dfrac{L(E,1)}{\Omega_E}$ can always be made a $p$-adic unit after "shifting by an isogeny". I wanted to ask:

1. Is this conjecture known to be true? It seems like if this is true, this would be well-known, but I'd like to confirm.

2. If it is true, does anyone have a reference?

Answer 1

The conjecture is False.

Let $\mathcal A$ be the isogeny class 6690j and $p=7$. The quantities $\dfrac{L(E,1)}{\Omega_E}$ for the two elliptic curves are $7$ and $49$ respectively, neither a $p$-adic unit.