Exponents of Class Groups and Elliptic Curves Original Research Article

We show that the number of elliptic curves over Q with conductor N is much less-thanvar epsilon N1/4+var epsilon, and for almost all positive integers N, this can be improved to much less-thanvar epsilon Nvar epsilon. 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(image) has size much less-thanvar epsilon D1/4+var epsilon, a non-trivial improvement over the Brauer–Siegel estimate.