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

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 $E/\mathbb{Q}$ 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.

New results on w4 (mod 5) and w6 (mod 7), non-brute-force proofs