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There are several upper bounds for number of elliptic curves (over $\mathbb{Q}$, 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 numbers (cube-free except for powers of 2 or 3, say) are not conductors. Or to be precise, is every possible candidate for conductor is actually a conductor ? Is there any result towards this ?

dragoboy
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  • What is the conductor of $y^2=x^3+ax+b$ with the 2,3 part removed (and assuming the discriminant is minimal) – reuns Jan 12 '20 at 07:56
  • 6th power (roughly) of conductor should be something lying in between disc and disc^6. To be more precise, it is related to reduction type. And I am asking, how often it is the case that it has bad reduction only at a given set of primes. For example, how often a prime conductor is possible ? equivalent to saying, how often 4a^3+27b^2 is prime. Isn't it hard ? May be a less harder would be to ask, how often set of prime factors of 4a^3+27b^2 comes from a pre-determined set. – dragoboy Jan 12 '20 at 08:11

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