1
$\begingroup$

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.

Thanks

$\endgroup$
2
$\begingroup$

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:

http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN365956996_0091

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

http://www.lmfdb.org/EllipticCurve/Q

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.