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A Complete classification of the rank of a certain family of elliptic curves

This paper provides a complete classification of the ranks for the family of elliptic curves $E_p: y^2 = x^3 - 6px$ (where $p$ is an odd prime), proving that the rank is 0 for $p=3$ , exactly 1 for all odd primes $p \not\equiv 1 \pmod{24}$ , and either 1 or 3 when $p \equiv 1 \pmod{24}$ .

Original authors: Richa Sharma, Arkabrata Ghosh

Published 2026-07-21

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CC BY 4.0

Original authors: Richa Sharma, Arkabrata Ghosh

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 a vast, invisible library where every book is a unique mathematical shape called an elliptic curve. These aren't just any shapes; they are the secret codebreakers of modern number theory, hiding deep patterns that help us understand how numbers behave. One of the most exciting mysteries about these curves is their "rank." Think of the rank as the number of independent keys needed to unlock the infinite doors of rational points on the curve. If the rank is zero, the door is locked tight. If it's one, there's one key. If it's three, you need a whole set of keys. Mathematicians have been trying to predict how many keys a specific curve will have for decades, but it's like trying to guess the weather in a different galaxy just by looking at the clouds. The question matters because these curves are the backbone of everything from secure internet encryption to the deepest unsolved puzzles in math.

In this paper, two mathematicians, Richa Sharma and Arkabrata Ghosh, decide to tackle a specific family of these curves, which they call $E_p$ . These curves follow a simple recipe: $y^2 = x^3 - 6px$ , where $p$ is a special kind of number called an odd prime (a number divisible only by 1 and itself, like 5, 7, or 11, but not 3). The authors act like detectives sorting through a massive pile of clues. They use a powerful mathematical tool called "2-descent," which is like a high-tech scanner that checks if a curve has hidden keys (points) or if it's completely empty. By scanning hundreds of different primes and looking at the "residue class" (a fancy way of saying the remainder you get when you divide the prime by 24), they manage to map out the entire landscape of this family.

Here is what they found: The rank of these curves is entirely determined by the prime number $p$ and how it behaves when divided by 24. If you pick a prime $p$ that leaves a remainder of 1 when divided by 24 (like 73 or 97), the curve is a bit of a mystery; it will have a rank of either 1 or 3. It's like rolling a die where you know the result is either a one or a three, but you can't tell which one just by looking at the prime. However, for every other type of odd prime (those leaving remainders of 5, 7, 11, 13, 17, 19, or 23), the answer is crystal clear: the rank is exactly 1. There are no surprises, no hidden three-key locks, just a single key every time.

The authors also checked a special case where the prime is 3. In this instance, the curve $y^2 = x^3 - 18x$ is completely locked down with a rank of 0, meaning it has no non-trivial rational points to speak of. While the result for the "remainder of 1" group relies on a famous but unproven guess in math called the "Parity Conjecture" to confirm that the rank must be odd (hence 1 or 3), the results for all other groups are proven facts. They didn't just guess; they used rigorous logic to prove that for almost all primes in this family, the complexity of the curve is perfectly predictable. It's a complete classification, turning a chaotic jungle of numbers into a neatly organized map where the only wild card is the specific behavior of primes that fit the "1 mod 24" pattern.

Technical Summary: A Complete Classification of the Rank of a Certain Family of Elliptic Curves

Problem Statement
This paper investigates the Mordell–Weil rank of the family of elliptic curves defined by the equation $E_p: y^2 = x^3 - 6px$ , where $p$ is an odd prime. While the rank of elliptic curves is a fundamental invariant in arithmetic geometry, determining it explicitly for specific families remains a challenging problem. The authors aim to provide a complete classification of the rank of $E_p(\mathbb{Q})$ based strictly on the residue class of the prime parameter $p$ modulo 24. The study also addresses the specific case where $p=3$ , resulting in the curve $E_3: y^2 = x^3 - 18x$ .

Methodology
The primary tool employed is the method of 2-descent, a classical technique used to bound the rank of elliptic curves possessing a rational point of order two. The authors utilize the following framework:

Isogeny and Descent Homomorphisms: The curve $E_p$ is analyzed alongside its 2-isogenous curve $\bar{E}_p: y^2 = x^3 + 24px$ . The authors define descent homomorphisms $\alpha: E_p(\mathbb{Q}) \to \mathbb{Q}^*/(\mathbb{Q}^*)^2$ and $\bar{\alpha}: \bar{E}_p(\mathbb{Q}) \to \mathbb{Q}^*/(\mathbb{Q}^*)^2$ .
Torsor Analysis: The images of these homomorphisms are determined by analyzing the solvability of associated homogeneous spaces (torsors) over the integers. These torsors take the form of Diophantine equations $N^2 = b_1 M^4 + b_2 e^4$ , where $b_1 b_2$ corresponds to the coefficients of the curve.
Local Solvability: The authors systematically test the solvability of these torsors by reducing them modulo various primes (specifically 3, 8, 16, and 24) and applying properties of Legendre symbols. This allows them to eliminate impossible divisors and identify the exact cardinality of the images $|\alpha(E_p(\mathbb{Q}))|$ and $|\bar{\alpha}(\bar{E}_p(\mathbb{Q}))|$ .
Rank Calculation: Using the formula $2^r = \frac{1}{4} |\alpha(E_p(\mathbb{Q}))| \cdot |\bar{\alpha}(\bar{E}_p(\mathbb{Q}))|$ , where $r$ is the rank, the authors derive upper bounds for the rank.
Parity Conjecture: To distinguish between odd ranks (1 and 3) when the upper bound is 3, the authors invoke the Parity Conjecture. They first calculate the global root number $w(E_p/\mathbb{Q})$ using local root number formulas derived by Birch and Stephens, finding $w(E_p/\mathbb{Q}) = -1$ for all odd primes $p$ . Under the assumption that the Parity Conjecture holds (or that the Tate–Shafarevich group is finite), this implies the rank must be odd.

Key Contributions and Results

Classification for $p \neq 3$ :
- Case $p \equiv 1 \pmod{24}$ : The rank is either 1 or 3. The 2-descent analysis shows the upper bound is 3. Assuming the Parity Conjecture, the rank is odd, hence 1 or 3.
- Case $p \not\equiv 1 \pmod{24}$ : For all other congruence classes of odd primes modulo 24 ( $p \equiv 5, 7, 11, 13, 17, 19, 23$ ), the rank is exactly 1. The 2-descent analysis yields an upper bound of 1, and the existence of a point of infinite order (implied by the non-triviality of the descent groups) confirms the rank is at least 1.
Case $p = 3$ : The authors prove that the curve $E_3: y^2 = x^3 - 18x$ has rank 0.
Conditional Results without Parity Conjecture: The paper provides specific conditions under which the rank is exactly 1 without relying on the Parity Conjecture:
- If $p \equiv 7 \pmod{24}$ and $(p-6)$ is a perfect square, the rank is 1.
- If $p \equiv 11, 17, 19 \pmod{24}$ and $(3p-2)$ is a perfect square, the rank is 1.

Significance
The paper claims to provide a complete classification of the ranks for the family $E_p$ based on the residue class of $p$ modulo 24. The results demonstrate a remarkably simple distribution of ranks governed entirely by modular arithmetic constraints on the prime parameter. Specifically, the family exhibits a dichotomy where the rank is almost always 1, except for the specific congruence class $p \equiv 1 \pmod{24}$ , where the rank may rise to 3. This work extends previous investigations into families of the form $y^2 = x^3 - Dx$ by addressing the specific coefficient structure $6p$ and utilizing 2-descent to resolve the rank for all odd primes. The findings are supported by computational examples verified using SAGE.

Technical Summary: A Complete Classification of the Rank of a Certain Family of Elliptic Curves

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