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We propose a new model for counting $\mathbf{GL}_2(\mathbb{F}_p)$-numberfields having the same local properties as the splitting field of the mod$p$-Galois representation associated with an elliptic curve over the rationalnumbers. We explain how this new model and Bhargava's local-to-globalheuristics for counting $\mathbf{GL}_2(\mathbb{F}_p)$-number fields both shedlight on the problem of estimating the number of elliptic curves over therational numbers of squarefree conductor $N < X.$  The new model predicts the existence of significantly more$\mathbf{GL}_2(\mathbb{F}_p)$-number fields with the desired local propertiesthan does the local-to-global model when $N$ has a large number of distinctprime factors, owing to degeneracies caused by Atkin-Lehner involutions. Wedescribe computational evidence supporting the new model.

On Bhargava's heuristics for $\mathbf{GL}_2(\mathbb{F}_p)$-number fields and the number of elliptic curves of bounded conductor