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.

Arithmetic of Elliptic Curvesbook. It says something much stronger than the fact that there are only finitely many elliptic curves of a given conductor. $\endgroup$ – Joe Silverman Sep 15 '12 at 22:08