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On group structures realized by elliptic curves over a fixed finite field

We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for the number of distinct group structures of all elliptic curves over a finite field. We use these formulas to derive some asymptotic estimates and tight upper and lower bounds for various counting functions related to classification of elliptic curves accordingly to their group structure. Finally, we present results of some numerical tests which exhibit several interesting phenomena in the distribution of group structures. We pose getting an explanation to these as an open problem.