TITLE: Finiteness of elliptic curves of a given conductor
QUESTION [10 upvotes]: 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.

REPLY [7 votes]: I just want to mention a couple of things to complement what Joe Silverman and Noam Elkies have already said.
As Brian Conrad pointed out in response to this ill-posed question of mine, you need the Shafarevich conjecture (for abelian varieties of all dimensions as proved by Faltings) to prove modularity of elliptic curves.
I also want to mention this paper of Cremona and Lingham:
Finding all elliptic curves with good reduction outside a given set of primes 
which deals with the algorithmic problems of finding all elliptic curves of a given conductor as discussed in Noam's first comment. They compute the $S$-integral points on $y^2 - x^3 = D$ assuming that a Mordell-Weil basis has been computed. The ideas are implemented in Sage and can be used to find the elliptic curves with everywhere good reduction outside a sufficiently simple finite set of primes $S$. 
For example, if you wanted to find all elliptic curves with everywhere good reduction outside $37$ you would use the following code
sage: egros = EllipticCurves_with_good_reduction_outside_S
sage: time curves = egros([37])
Time: CPU 1.49 s, Wall: 1.50 s
sage: for E in curves:
....:     print E.a_invariants()
....:     
(0, 0, 1, -1, 0)
(0, 1, 1, -23, -50)
(0, 1, 1, -1873, -31833)
(0, 1, 1, -3, 1)
(0, 1, 1, -4563, 116200)
(0, 1, 1, -31943, -2138543)
(0, 1, 1, -2564593, -1581651042)
(1, -1, 0, 3166, -59359)
(1, -1, 0, -2276219, -1321241558)
(1, -1, 1, 2, -2)
(1, -1, 1, -1663, -25680)
(0, 0, 1, -1369, 12663)
(0, 1, 1, -12, -17)
(0, 1, 1, -2602, 50229)
(0, 1, 1, -16884, -647702)
(0, 1, 1, -3562594, 2587011456)