There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is finite.
One way to prove this, which Cremona and Lingham use here to compute all such curves, is to use Siegel's theorem that an elliptic curve over $Q$ has only a finite number of $S$-integral points.
Here's a proof with overkill:
Given $S$ there are a finite number of possible conductors $N$ for elliptic curves with everywhere good reduction outside $S$. They must all be divisors of $2^8 3^5 d^2$ where $d$ is the product of those primes in $S$ different from 2 and 3.
The corresponding spaces $S_2(\Gamma_0(N))$ of cuspforms for each of our finite list of $N$ is finite dimensional.
By the modularity theorem, there is hence finite number isogeny classes of elliptic curves with everywhere good reduction outside $S$.
By Mazur's Modular Curves and the Eisenstein Ideal there are only a finite number of isomorphism classes of elliptic curves in a given isogeny class.
Question 1: Does any of this machinery rely on Siegel's theorem?
Question 2: If the answer to question 1 is no, can this proof of Shafarevich's theorem be "cheaply extended" to deduce Siegel's Theorem from these seemingly unrelated powerful results?
By "cheaply extended" I mean without the use of techniques with the diophantine flavor of Baker's theory of linear forms in logarithms.
