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The moduli space $M_1(X, e)$ of degree $e$ elliptic curves on $X$ has virtual dimension zero if $X$ is a Calabi-Yau 3-fold. I am wondering if there is an example of such an $X$ so that each $M_1(X, e)$ is finite and moreover all but finitely many are empty. Are there examples where all $M_1(X, e)$ are finite and nonempty?

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