Abstract
We show that the number of elliptic curves over Q with conductor N is ⪡εN1/4+ε, and for almost all positive integers N, this can be improved to ⪡εNε. The second estimate follows from a theorem of Davenpart and Heilbronn on the average size of the 3-class groups of quadratic fields. The first estimate follows from the fact that the 3-class group of a quadratic field Q(D) has size ⪡ε|D|1/4+ε, a non-trivial improvement over the Brauer–Siegel estimate.
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