PreprintCC BY 4.0arXiv
Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves
Abstract
We investigate the extent to which the coefficients (w1,w2,w3,w4,w6) of the reduced minimal Weierstrass model of an elliptic curve E/ℚ 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.
Related
- On the finite transcendence of Frobenius traces for abelian varieties over ℚ
- Integer points for iterated Weierstrass elliptic functions
- The L1-realization of an elliptic operator with unbounded coefficients
- Sharp maximum estimates for linear uniformly elliptic equations with measurable coefficients
- Uniform estimates for elliptic systems with critical Sobolev coefficients
BibTeX
@misc{banwait2026decision,
title = {{Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves}},
author = {Barinder S. Banwait and Xiaoyu Huang and Kyu-Hwan Lee and Seewoo Lee and Thomas Oliver and Alexey Pozdnyakov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.24251},
eprint = {2607.24251},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.24251},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves
AU - Barinder S. Banwait
AU - Xiaoyu Huang
AU - Kyu-Hwan Lee
AU - Seewoo Lee
AU - Thomas Oliver
AU - Alexey Pozdnyakov
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2607.24251
UR - https://arxiv.org/abs/2607.24251
AB - We investigate the extent to which the coefficients (w_1,w_2,w_3,w_4,w_6) of the reduced minimal Weierstrass model of an elliptic curve E/ℚ are determined by the Dirichlet coefficients a_n(E) of its L-function, whose values at primes of good reduction are the Frobenius traces of E. We prove that w_1, w_2 and w_3 are given by explicit formulae in a_2(E), a_3(E) and a_4(E), that w_4 modulo $5$ is then determined by a_5(E), and that w_6 modulo $7$ is determined by a_7(E) together with w_1,w_2,w_3,w_4. 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.
ER -