I am looking for an example of a prime, p, for which there exists two $p$-ordinary rational elliptic curves $E$, $F$ for which, at every prime $l$ not dividing $N=p \operatorname{Cond}(E) \operatorname{Cond}(F)$:

$$\#E(\mathbb{F}_l) \equiv \#E( \mathbb{F}_l) \mod p $$

Such a congruence could be detected by a Hida Family having two weight $2$ modular forms $f,g,$ whose fields of fourier coefficients $K_f=K_g=\mathbb{Q}$. What I would prefer is exact equations of $E$ and $F$ (say, on www.lmfdb.org).

I am looking for an example of a prime, p, for which there exists two $p$-ordinary rational elliptic curves $E$, $F$ for which, at every prime $l$ not dividing $N=p \operatorname{Cond}(E) \operatorname{Cond}(F)$:

$$\#E(\mathbb{F}_l) \equiv \#F( \mathbb{F}_l) \mod p $$

Such a congruence could be detected by a Hida Family having two weight $2$ modular forms $f,g,$ whose fields of fourier coefficients $K_f=K_g=\mathbb{Q}$. What I would prefer is exact equations of $E$ and $F$ (say, on www.lmfdb.org).