# Conditions such that $\# E_{\mathrm{ns}}(\mathbb{F}_{\ell})\not\equiv 0 \bmod{p}$

Let $E/\mathbb{Q}$ be a semistable elliptic curve. Let $\ell$ be a prime of multiplicative reduction and consider $\# E_{\mathrm{ns}}(\mathbb{F}_{\ell})$. Given a prime $p \neq \ell$, are there any restrictions that I can put on $E$ which force $\# E_{\mathrm{ns}}(\mathbb{F}_{\ell})\not\equiv 0 \bmod{p}$?

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You can calculate the size of $E_\text{ns}(\mathbb{F}_\ell)$. See Chapter III, Exercise 3.5 of Silverman's "The Arithmetic of Elliptic Curves".