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 ?