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). 



Example of two p-Ordinary Elliptic Curves congruent to each other
There are many such curves. For $p = 2$, one can just take two curves $E$ and $F$ with $E : y^{2} = f(x)$ and $F : y^{2} = g(x)$ where $f(x)$ and $g(x)$ define the same number field. Also, if $E$ and $F$ are isogenous, then the above statement is true for any $p$ (but the corresponding modular forms are the same then).


Rubin and Silverberg have a couple of papers (see [the MathSciNet reference here](https://mathscinet.ams.org/mathscinet-getitem?mr=1363500)) that handle $p = 3$ and $5$. In particular, for these $p$, given an elliptic curve $E/\mathbb{Q}$, they construct a one-parameter family of elliptic curves $E\_{t}$ so that $E[p]$ and $E\_{t}[p]$ are isomorphic as Galois modules. (This works because such $E\_{t}$ are parametrized by a twist of $X(p)$, and $X(p)$ has genus zero for $p = 3$ and $5$).


For larger $p$, there are still pairs (but not quite as many), and the best place to look is in [this paper of Tom Fisher](https://www.dpmms.cam.ac.uk/~taf1000/papers/highercongr.pdf). In particular, the largest known example is with $p = 17$, which I mentioned in my answer to the question [here](https://mathoverflow.net/questions/269035/elliptic-curves-with-the-same-mod-p-representation/269057#269057). This example is a pair of $17$-ordinary elliptic curves that have $\# E(\mathbb{F}\_{\ell}) \equiv \# F(\mathbb{F}\_{\ell}) \pmod{17}$ for all primes $\ell$.


