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There are several upper bounds for number of elliptic curves (over Q, say) upto-isomorphism with a given conductor N. Probably the best one is given by Helfgott-Venkatesh of order N^{0.22} (or may be some improvement is possible knowing improved bounds for 3-torsion in class groups).

My question is whether there is any lower bound ? I do not how much of this question make sense because, it feels like most of the possible candidate for conductor is not a conductor. Is there a related result ?

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  • $\begingroup$ Naively like this the lower bound must be zero. For all number divisible by $5^3$ there cannot be an elliptic curve of that conductor. Similar for all $p^3$ with $p\geq 5$ and sufficiently high powers of $2$ and $3$, too. $\endgroup$
    – Chris Wuthrich
    Commented Jan 12, 2020 at 20:19
  • $\begingroup$ This is why I said, whether "all possible candidates" are conductor. I know from definition $p^3$ is not a conductor. By possible candidates, I really meant whether all cube free (forget powers of 2,3). I hope my question is clear now ? $\endgroup$
    – dragoboy
    Commented Jan 12, 2020 at 20:26
  • $\begingroup$ Well, $N=2$ is a famous example for which there are no curves. In the current Cremona tables of all curves with $N$ below half a million, there are plenty of $N$ for which there are no curve or just one. Among the 303957 squarefree $N$ there are 149006 without a curve and 52436 with just one curve. $\endgroup$
    – Chris Wuthrich
    Commented Jan 13, 2020 at 9:56
  • $\begingroup$ That sounds interesting. Is density of such set known to be zero ? (again, I mean density in the set of cube free numbers). I think I saw some conjecture like that, but unable to find the reference now. $\endgroup$
    – dragoboy
    Commented Jan 13, 2020 at 10:55
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    $\begingroup$ It is conjectured that there are roughly $X^{5/6}$ elliptic curves with conductor less than $X$. This would imply density zero. The corresponding result for heights is due to Bhargava-Shankar. Arul Shankar has thought about the conjecture for conductors quite a lot. If I wanted to know the state of the art about this question, I would probably read this paper (arxiv.org/pdf/1904.13063.pdf) and then if I still had questions, I would ask Arul. $\endgroup$
    – James Weigandt
    Commented Jan 19, 2020 at 15:10

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