I'm wondering if the following conjecture is true:



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> 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.
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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?



$p$-adic valuation of $L$ values for elliptic curves
The conjecture is **False**.


Let $\mathcal A$ be the isogeny class [6690j](https://www.lmfdb.org/EllipticCurve/Q/6690/j/) and $p=7$. The quantities $\dfrac{L(E,1)}{\Omega\_E}$ for the two [elliptic](https://www.lmfdb.org/EllipticCurve/Q/6690/j/2) [curves](https://www.lmfdb.org/EllipticCurve/Q/6690/j/1) are $7$ and $49$ respectively, neither a $p$-adic unit.


