Abstract
It is proved that the 3-part of the class number of a quadratic field is in general and if has a divisor of size . These bounds follow as results of nontrivial estimates for the number of solutions to the congruence modulo in the ranges and , where are nonzero integers and is a square-free positive integer. Furthermore, it is shown that the number of elliptic curves over with conductor is in general and if has a divisor of size . These results are the first improvements to the trivial bound and the resulting bound for the 3-part and the number of elliptic curves, respectively.
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