Counting elliptic curves over $\mathbb{Q}$ with bounded naive height

In this paper, we give exact and asymptotic formulas for counting elliptic curves $E_{A,B}\text{\hspace{0.17em}⁣}:y^2=x^3+Ax+B$ with $A,B\in \mathbb{Z}$, ordered by naive height. We study the family of all such curves and also several natural subfamilies, including those with fixed $j$-invariant and those with complex multiplication (CM). In particular, we provide formulas for two commonly used normalizations of the naive height appearing in the literature: the calibrated naive height, defined by $$H^{\mathrm{c}\mathrm{a}\mathrm{l}}(E_{A,B}):=\max \{4\mathrm{\mid }A{\mathrm{\mid }}^3,27B^2\},$$ and the uncalibrated naive height, defined by $$H^{\mathrm{n}\mathrm{c}\mathrm{a}\mathrm{l}}(E_{A,B}):=\max \{\mathrm{\mid }A{\mathrm{\mid }}^3,B^2\}\mathrm{.}$$ In fact, we prove our theorems with respect to the more general naive height $H_{\alpha ,\beta }(E_{A,B}):=\max \{\alpha \mathrm{\mid }A{\mathrm{\mid }}^3,\beta B^2\}$, defined for arbitrary positive real numbers $\alpha ,\beta \in {\mathbb{R}}_{>0}$. As part of our approach, we give a completely explicit parametrization of the set of curves $E_{A,B}$ with fixed $j$-invariant and bounded naive height, describing them as twists of the curve $E_{A_j,B_j}$ of minimal naive height for the given $j$-invariant. We also include tables comparing and verifying our theoretical predictions with exact counts obtained via exhaustive computer searches, and we compute data for CM elliptic curves of naive height up to ${10}^{30}$. Code in SageMath is provided to compute all exact and asymptotic formulas appearing in the paper.