# Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves

arXiv: https://arxiv.org/abs/2607.24251 · HTML: https://mlsanity.com/paper/2607.24251/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 21 references that are in the map.
- First posted on arXiv 27 Jul 2026.

## Builds on (most cited first, 10 of 10)

- [Machine-Learning the Sato--Tate Conjecture](https://mlsanity.com/paper/2010.01213/) (2020)
- [Murmurations of elliptic curves](https://mlsanity.com/paper/2204.10140/) (2022)
- [Machine-Learning Arithmetic Curves](https://mlsanity.com/paper/2012.04084/) (2020)
- [Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer](https://mlsanity.com/paper/1911.02008/) (2019)
- [Machine Learning Class Numbers of Real Quadratic Fields](https://mlsanity.com/paper/2209.09283/) (2022)
- [Learning Euler Factors of Elliptic Curves](https://mlsanity.com/paper/2502.10357/) (2025)
- [Machine Learning Approaches to the Shafarevich-Tate Group of Elliptic Curves](https://mlsanity.com/paper/2412.18576/) (2024)
- [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)
- [Learning Fricke signs from Maass form Coefficients](https://mlsanity.com/paper/2501.02105/) (2025)

## Abstract (arXiv)

We investigate the extent to which the coefficients (w1,w2,w3,w4,w6) of the reduced minimal Weierstrass model of an elliptic curve E/mathbbQ are determined by the Dirichlet coefficients an(E) of its L-function, whose values at primes of good reduction are the Frobenius traces of E. We prove that w1, w2 and w3 are given by explicit formulae in a2(E), a3(E) and a4(E), that w4 modulo 5 is then determined by a5(E), and that w6 modulo 7 is determined by a7(E) together with w1,w2,w3,w4. These formulae, which appear to be new, were discovered by training decision tree models on the LMFDB; we report the accompanying experiments and explore applications to computing tables of elliptic curves.