What are some useful facts/algorithms for elliptic curves that can be obtained (proved completely) using the theory of modular forms without heavy machinery?
It's often been asked what elementary applications of modular forms are (partition congruences, counting solutions to quadratic forms...) and while these are all interesting, to me they don't capture the most compelling number theoretic reason we study modular forms, which is to apply them to elliptic curves (and Galois representations, but that's a different question).
Modular forms are essential to the theory of elliptic curves, but the typical applications rely on very deep theorems (Fermat's last theorem, modularity theorem, Birch-and-Swinnerton Dyer Conjecture, Sato-Tate...). Can we obtain some concrete number theoretic information about elliptic curves without relying on these deep theorems?
Here is an example of what I am looking for. The theory of elliptic functions is useful for elliptic curves because, for instance, they describe all isogenies between elliptic curves over subfields of $\mathbb C$, and can also be used to compute the isogenies explicitly, by writing $\wp (\alpha z)$ as a rational function of $\wp (z)$. This gives us very concrete information.
Treat the word "elementary" relatively--probably nothing here is going to be easy. In particular, we can't say anything about all elliptic curves; we can only say things about particular elliptic curves. One first has to find an elliptic curve associated to a modular form before one can get information, and so probably has to invoke Eichler-Shimura in some form. There are, however, examples where one can compute the elliptic curve associated to a modular form relatively easily. For example, in Knapp's Elliptic Curves he computes that the elliptic curve associated with the eigenform $f\in S_2(\Gamma_0(11))$ is $E: y^2+y=x^3-x^2-10x-20$, and there is a map $X_0(11)\to E(\mathbb C)$. (This is easy because $S_2(\Gamma_0(11))$ is 1-dimensional with genus 1.)
However, after we know that that a specific elliptic curve corresponds to a modular form, what can we concretely do with this knowledge? Saying their $L$-functions correspond is nice but to me, not the type of concrete result I'm looking for.