I'm looking for some references regarding the above topic.

To be more specific, references that address questions such as

  1. Given $D > 0$, how many elliptic curves over $\mathbb{Q}$ are there with (minimal) discriminant $< D$?

  2. Alternatively (and of course relatedly), given $A, B > 0$, how many elliptic curves are there that have in their minimal Weierstrass equation $|a| < A, |b| < B$?

  3. The same questions except considering the conductor instead of discriminant.



I believe that the standard reference is

Silverman and Brumer, The number of elliptic curves over $\mathbb{Q}$ with conductor $N$, Manuscripta Mathematica 91, 1996.

They prove that the number of elliptic curves of curves of conductor $N$ is bounded above by $N^{\frac{1}{2}+\epsilon}$.

I found out that you can read it here:


John Cremona's tables will give you the curves of a given conductor $N$ for all $N < 300,000$ and is available here:



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