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

This paper establishes exact and asymptotic formulas for counting elliptic curves over $\mathbb{Q}$ with bounded naive height, covering the general family and specific subfamilies like those with fixed $j$ -invariants or complex multiplication, while providing explicit parametrizations, computational verification, and SageMath code for various height normalizations.

Original authors: Adrian Barquero-Sanchez, Daniel Mora-Mora

Published 2026-06-08

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CC BY 4.0

Original authors: Adrian Barquero-Sanchez, Daniel Mora-Mora

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the world of numbers as a vast, infinite library. Inside this library, there are millions of special books called elliptic curves. These aren't stories about dragons or space travel; they are mathematical objects defined by a specific equation: $y^2 = x^3 + Ax + B$ .

In this paper, the authors, Adrian Barquero-Sánchez and Daniel Mora-Mora, act as librarians trying to count how many of these books exist, but with a specific rule: they only want to count the books where the numbers $A$ and $B$ aren't too huge.

Here is a breakdown of their work using simple analogies:

1. The "Naive Height" (The Size of the Book)

To decide which books to count, the librarians need a way to measure the "size" of a curve. They use something called naive height.

The Analogy: Imagine every curve has a "weight" based on how big its numbers $A$ and $B$ are. The "naive height" is like a scale that tells you the heaviest number in the equation.
Two Ways to Weigh: The paper discusses two different ways to set up this scale:
- The "Uncalibrated" Scale: A simple, rough measurement (just looking at the raw numbers).
- The "Calibrated" Scale: A more precise, adjusted measurement that mathematicians have found to be better for certain deep theories.
The Goal: The authors want to know: "If I set a weight limit of $X$ , how many curves are lighter than that?"

2. The Main Discovery: Exact Formulas

Before this paper, mathematicians had rough guesses (asymptotic formulas) for how many curves fit under a weight limit. They knew the general shape of the answer, but not the exact details.

The Breakthrough: The authors created exact recipes (formulas) to count these curves. It's like going from saying, "There are probably about a million apples in the orchard," to having a formula that says, "There are exactly 1,042,399 apples, plus or minus a tiny error."
They did this for:
- All curves: The entire library.
- Curves with a specific "ID": Every curve has a unique fingerprint called a $j$ -invariant. The authors figured out how to count curves that share the same fingerprint.
- Curves with "Special Powers" (CM): Some curves have a special property called Complex Multiplication (CM). These are like rare, magical editions of the books. The authors counted exactly how many of these rare books exist under a certain weight.

3. The "Twist" and the "Minimal Model"

One of the most interesting parts of the paper is how they organize the curves.

The Analogy: Imagine you have a perfect, small toy car (the minimal model). You can make bigger, heavier versions of this car by stretching it (a twist).
The Finding: The authors discovered that for any specific "fingerprint" ( $j$ -invariant), every single curve in the library is just a "stretched version" (a twist) of one of two tiny, minimal cars.
Why it matters: Instead of searching for millions of random curves, you only need to find these two tiny "seed" curves. Once you have them, you can generate the entire family of curves with that fingerprint by applying a simple mathematical stretching rule. This makes counting them incredibly easy.

4. The "Missing" Magical Curves

The authors used their new formulas to solve a mystery.

The Mystery: In a previous computer search, mathematicians looked for "magical" curves (CM curves) with a specific, very complex fingerprint (related to the number -163) but found none up to a certain size.
The Explanation: The authors proved theoretically that these curves do exist, but they are enormously heavy.
The Result: Their formula showed that the smallest version of these specific curves has a weight so massive (around $10^{21}$ ) that it was impossible for the previous computer search to find them. It's like looking for a needle in a haystack, but the needle is actually a giant steel beam hidden under a mountain. The paper explains why it wasn't found: it was simply too big for the search range.

5. The "Density" of the Library

Finally, the authors looked at the ratio of "unique" books to "duplicate" books.

The Finding: They calculated that about 99.9% of the curves in their "uncalibrated" list are actually unique representatives. The other 0.1% are just duplicates (twists) of the same underlying shape.
The Takeaway: If you pick a random curve from the library, it is almost certainly a unique shape, not a copy of another.

Summary

In short, this paper is a master catalog for elliptic curves. The authors:

Created precise formulas to count how many curves exist below a certain size.
Showed that all curves with a specific identity are just "stretched" versions of two tiny "seed" curves.
Explained why certain rare, "magical" curves were missing from previous computer searches (because they are astronomically large).
Provided the code and data so anyone can verify these counts, effectively turning a guessing game into a precise science.

Technical Summary: Counting Elliptic Curves Over $\mathbb{Q}$ with Bounded Naive Height

Problem Statement
This paper addresses the problem of counting elliptic curves over the rational numbers $\mathbb{Q}$ , ordered by their "naive height." Specifically, the authors consider curves in short Weierstrass form $E_{A,B}: y^2 = x^3 + Ax + B$ with integer coefficients $A, B \in \mathbb{Z}$ . The study focuses on two primary normalizations of the naive height widely used in arithmetic statistics: the calibrated naive height, $H_{\text{cal}}(E_{A,B}) := \max\{4|A|^3, 27B^2\}$ , and the uncalibrated naive height, $H_{\text{ncal}}(E_{A,B}) := \max\{|A|^3, B^2\}$ .

The authors aim to provide exact and asymptotic formulas for the number of such curves within a height bound $X$ across several families:

The global family of all such curves (both minimal models and non-minimal models).
Subfamilies with a fixed $j$ -invariant.
Subfamilies of curves with Complex Multiplication (CM).

A specific challenge addressed is the distribution of curves with fixed $j$ -invariants where $j \in \mathbb{Q} \setminus \{0, 1728\}$ . Previous work (e.g., [BSCM25]) had established asymptotic formulas for $j=0$ and $j=1728$ but could only provide an upper bound of $O(X^{1/6})$ for other rational $j$ -invariants.

Methodology
The authors unify the treatment of calibrated and uncalibrated heights by introducing a generalized naive height $H_{\alpha,\beta}(E_{A,B}) := \max\{\alpha|A|^3, \beta B^2\}$ for arbitrary real parameters $\alpha, \beta \ge 1$ .

The core methodological contributions involve:

Parametrization of Fixed $j$ -Invariant Families: The authors derive a completely explicit parametrization for the family of curves $E_{A,B}$ with a fixed $j$ -invariant $j \in \mathbb{Q} \setminus \{0, 1728\}$ . They show that these curves correspond to integer points on a cuspidal cubic curve $y^2 = a(j)x^3$ , where $a(j) = \frac{4(1728-j)}{27j}$ .
Integral Points on Cuspidal Cubics: A key technical tool is Proposition 3.1, which provides an exact formula for the number of integral points on a cuspidal cubic within a rectangular box. This relies on a bijection between the set of integral points and a scaled set of integers $m \in \mathbb{Z} \setminus \{0\}$ .
Twist Decomposition: The authors utilize the decomposition of the family of all curves $\tilde{E}$ into twists of minimal models $E$ . Specifically, any curve $E_{A',B'}$ can be uniquely written as $d * E_{A,B}$ (where $d$ is an integer and $E_{A,B}$ is a minimal model) such that the height scales by $d^{12}$ . This allows the use of Möbius inversion to relate the count of all curves to the count of minimal models (isomorphism classes).
Asymptotic Analysis: By combining the exact counts of integral points with the density of square-free integers (via the Möbius function) and $k$ -free integers, the authors derive precise asymptotic expansions, including error terms.

Key Contributions and Results

Global Counting Formulas: The paper provides exact and asymptotic formulas for the total number of curves in the families $\tilde{E}$ (all integral models) and $E$ (minimal models) ordered by $H_{\alpha,\beta}$ .
- For minimal models, the asymptotic count is $\frac{4}{\alpha^{1/3}\beta^{1/2}\zeta(10)}X^{5/6} + O(X^{1/2})$ .
- This unifies and extends previous results by Brumer (for uncalibrated height) and Poonen/Balakrishnan et al. (for calibrated height).
Fixed $j$ -Invariant Formulas: The authors provide the first explicit asymptotic main term for curves with a fixed $j$ -invariant $j \in \mathbb{Q} \setminus \{0, 1728\}$ .
- They prove that the number of minimal models is $\frac{2c(j; H_{\alpha,\beta})}{\zeta(2)}X^{1/6} + O(X^{1/12})$ , where $c(j; H_{\alpha,\beta})$ is an explicit constant depending on $j$ and the height normalization.
- This improves upon the previous $O(X^{1/6})$ bound, providing a computable leading coefficient.
- For $j=0$ and $j=1728$ , the formulas are refined to include explicit constants and error terms $O(X^{1/12})$ .
Complex Multiplication (CM) Counts: By summing the results over the thirteen known CM $j$ -invariants, the authors derive exact and asymptotic formulas for the number of CM elliptic curves.
- The asymptotic expansion for minimal CM models includes three terms: $O(X^{1/2})$ (from $j=0$ ), $O(X^{1/3})$ (from $j=1728$ ), and $O(X^{1/6})$ (from other CM $j$ -invariants).
- The paper explicitly calculates the constant for the $X^{1/6}$ term, denoted $K(\alpha, \beta)$ .
Explicit Parametrization and Minimal Models: The paper provides a constructive parametrization $A(m), B(m)$ for all curves with a fixed $j$ -invariant. This parametrization identifies the two curves of minimal naive height (corresponding to $m = \pm 1$ ) and describes all other curves in the family as quadratic twists of these minimal models.
- Example: For the CM order with discriminant $-163$, the minimal height curves have a calibrated height of approximately $2.63 \times 10^{21}$ . This explains why previous exhaustive computer searches up to $10^{10}$ found no such curves.
Asymptotic Density: The paper calculates the asymptotic density of minimal models within the set of all integral models.
- For the global family, the density is $1/\zeta(10) \approx 99.9\%$ .
- For fixed $j$ -invariants ( $j \neq 0, 1728$ ), the density is $1/\zeta(2) \approx 60.8\%$ .

Verification and Data
The authors verify their theoretical formulas against exhaustive computer searches for small height bounds. They also provide extensive numerical data for CM curves up to a height of $10^{30}$ , computed using their exact formulas rather than brute-force enumeration. This data includes tables of minimal height representatives for all thirteen CM $j$ -invariants and counts of curves for various height bounds.

Significance
The paper claims significance in providing a unified framework for counting elliptic curves under different height normalizations. Its primary theoretical advance is the derivation of explicit asymptotic formulas for curves with fixed $j$ -invariants (excluding 0 and 1728), a problem previously limited to upper bounds. Furthermore, the explicit parametrization of fixed $j$ -invariant families allows for the identification of minimal models and explains discrepancies between theoretical predictions and computational searches for high-height CM curves. The inclusion of SageMath code and extensive numerical tables serves as a computational complement to the theoretical results, facilitating further research in arithmetic statistics.