Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves
math.NT, cs.LG
Submitted: 2026-07-27
Updated: 2026-09-18
Comments: New results on w4 (mod 5) and w6 (mod 7), non-brute-force proofs
Code: https://github.com/cocoxhuang/int2int2
License: http://creativecommons.org/licenses/by/4.0/
The gist: 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
Terminology
Abstract
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.