It follows from the modularity theorem for elliptic curves over $\mathbb{Q}$ that there are finitely many elliptic curves of a given conductor $N$. Moreover, one can algorithmically enumerate them. [Edit: As Emerton comments below, without further argument, this is only true for elliptic curves up to isogeny!]
Was the finiteness and/or algorithmic enumeration of elliptic curves of a given conductor known before the modularity theorem?
Every elliptic curve over $\mathbb{Q}$ can be written in the form $y^2 = x^3 + ax + b$ where $a, b \in \mathbb{Z}$ with discriminant $\Delta = -16(4a^3 +27b^2) \neq 0$. So the number of elliptic curves of discriminant $D$ is bounded above by number of nontrivial pairs $(a, b) \in \mathbb{Z}^2$ such that $D = -16(4a^3 +27b^2)$.
Let $D \in{\mathbb{Z}}, D \neq 0$ be given. Because $D \neq 0$, the cubic equation $b^2 = \frac{-D}{16\cdot27} – \frac{4a^3}{27}$ is nonsingular, so by Siegel's theorem there are finitely many solutions. It follows that there are finitely many elliptic curves of a given discriminant. Silverman's book says that Baker even gave an explicit upper bound in this case, which was refined by Stark.
However, a priori there is no bound on the size of the discriminant of elliptic curves of a given conductor, so it doesn't immediately follow that there are finitely many elliptic curves of a given conductor. Szpiro's conjecture implies that if one fixes the conductor there are only finitely many discriminants that give that conductor. However, this conjecture is open (or not, depending on the status of Mochizuki's work).
Is there a weaker form of Szpiro's conjecture that has been proved giving an upper bound on the discriminant of an elliptic curve of a given conductor? If so, what's the minimum amount of technology needed to get the results?
Of course there are also issues of effectivity as well, which I also welcome comments on.
Best Answer
As Noam says, this was well-known long before Wiles. The proof that he sketches can be used to prove Shafarevich's theorem for elliptic curves, i.e., given a finite set of primes $S$, there are only finitely many elliptic curves over $\mathbb{Q}$ with good reduction outside of $S$. So if you have a particular $N$ in mind, you can take $S$ to be the set of primes dividing $N$.
And there were even some fairly explicit upper bounds for the number of elliptic curves of conductor $N$. See for example my paper with Brumer, "The number of elliptic curves over $\mathbb{Q}$ with conductor $N$," Manuscripta Math. 91 (1996), 95-102. Turning things around, we can then use Wiles' theorem and the previously proven upper bound for the number of elliptic curves of conductor $N$ to immediately deduce a non-trivial upper bound for the number of elliptic factors of $J_0(N)$.