# Murmurations of elliptic curves

arXiv: https://arxiv.org/abs/2204.10140 · HTML: https://mlsanity.com/paper/2204.10140/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 24 references that are in the map.
- First posted on arXiv 21 Apr 2022.

## Builds on (most cited first, 3 of 3)

- [Machine-Learning the Sato--Tate Conjecture](https://mlsanity.com/paper/2010.01213/) (2020)
- [Machine-Learning Arithmetic Curves](https://mlsanity.com/paper/2012.04084/) (2020)
- [Machine-Learning Number Fields](https://mlsanity.com/paper/2011.08958/) (2020)

## Built on by (most cited first, 14 of 14)

- [Machine learning detects terminal singularities](https://mlsanity.com/paper/2310.20458/) (2023)
- [Machine Learning Class Numbers of Real Quadratic Fields](https://mlsanity.com/paper/2209.09283/) (2022)
- [Machine Learning Clifford invariants of ADE Coxeter elements](https://mlsanity.com/paper/2310.00041/) (2023)
- [A Triumvirate of AI Driven Theoretical Discovery](https://mlsanity.com/paper/2405.19973/) (2024)
- [From Black Box to Bijection: Interpreting Machine Learning to Build a Zeta Map Algorithm](https://mlsanity.com/paper/2511.12421/) (2025)
- [Flow-based Extremal Mathematical Structure Discovery](https://mlsanity.com/paper/2601.18005/) (2026)
- [Machines Learn Number Fields, But How? The Case of Galois Groups](https://mlsanity.com/paper/2508.06670/) (2025)
- [Murmurations, Mestre--Nagao sums, and Convolutional Neural Networks for elliptic curves](https://mlsanity.com/paper/2603.17681/) (2026)
- [Machine Learning Mutation-Acyclicity of Quivers](https://mlsanity.com/paper/2411.04209/) (2024)
- [Learning Fricke signs from Maass form Coefficients](https://mlsanity.com/paper/2501.02105/) (2025)
- [Notes on a Path to AI Assistance in Mathematical Reasoning](https://mlsanity.com/paper/2310.02896/) (2023)
- [Learning Modular Exponentiation with Transformers](https://mlsanity.com/paper/2506.23679/) (2025)
- [Murmurations: a case study in AI-assisted mathematics](https://mlsanity.com/paper/2603.09680/) (2026)
- [Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves](https://mlsanity.com/paper/2607.24251/) (2026)

## Abstract (arXiv)

We investigate the average value of the Frobenius trace at p over elliptic curves in a fixed conductor range with given rank. Plotting this average as p varies over the primes yields a striking oscillating pattern, the details of which vary with the rank. Based on this observation, we perform various data-scientific experiments with the goal of classifying elliptic curves according to their ranks.