Very recently, I made an observation from scanning lists of elliptic curves on the LMFDB that leads me to the following (unproven) statement:

Fix $\ell \in \{5, 7\}$. Let $E$ be an elliptic curve over $\mathbb{Q}$ with a rational $\ell$-torsion generator. Then if $p \neq \ell$ is a prime at which $E$ has multiplicative reduction, and if $p$ appears in the prime factorization of the denominator of $j(E)$ with exponent not divisible by $\ell$, we have $p \equiv \pm 1$ (mod $\ell$).

I don't know how to get a spreadsheet of $j$-invariants of elliptic curves satisfying the hypothesis given above so I couldn't check this efficiently, but this was the case for each of the dozens of elliptic curves I checked. I noticed other strong trends as well, such as the fact that primes $p \neq \ell$ appearing in the factorization of the \textit{numerator} of $j(E)$ with exponent not a multiple of $\ell$ tend to satisfy $p \equiv \pm 1$ (mod $\ell$), but this is not true all of the time -- the exceptions are always smaller primes and always appear with exponent $3$ in the data I checked. Also, such a statement fails badly when I loosen the assumption about a rational $\ell$-torsion point to requiring only that the mod-$\ell$ Galois image be a Borel subgroup.

Is there any obvious proof or reason to expect that the main assertion above is true? Right now I don't see any way to prove it using rational points on modular curves, but I'm really not an expert on modular curves. No strategy using Neron models is jumping out at me either.