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

Abstract

In this paper, we give exact and asymptotic formulas for counting elliptic curves $ E_{A,B} \colon 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 \[ H^{\mathrm{cal}}(E_{A,B}) := \max\{ 4|A|^3, 27B^2 \}, \] and the uncalibrated naive height, defined by \[ H^{\mathrm{ncal}}(E_{A,B}) := \max\{ |A|^3, B^2 \}. \] In fact, we prove our theorems with respect to the more general naive height $H_{\alpha, \beta}(E_{A,B}) := \max\{ \alpha |A|^3, \beta B^2 \}$, 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.