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

2026-07-27

AI Summary

This work addresses the problem of efficiently recovering the first three reduced minimal Weierstrass coefficients of an elliptic curve from the Frobenius traces at primes 2 and 3 together with the parity of its conductor. The authors present and rigorously prove, for the first time, a set of explicit formulas that enable exact computation of these coefficients, demonstrating that they are uniquely determined by the isogeny class of the elliptic curve. By integrating tools from algebraic number theory—specifically conductor theory and Weierstrass equation formalism—with machine learning techniques, notably decision tree models, the method achieves perfect reconstruction accuracy. This establishes a clear and deterministic relationship among Frobenius traces, conductor parity, and the isogeny class.

Abstract

We investigate the extent to which the reduced minimal Weierstrass coefficients of an elliptic curve over $\mathbb{Q}$ may be computed from it's Frobenius traces. Decision tree models reveal that the first two reduced minimal Weierstrass coefficients can be recovered with perfect accuracy from the Frobenius traces at the primes $2$ and $3$, and the third by supplementing these two traces with the conductor parity. We subsequently prove explicit formulae for these coefficients using the Frobenius traces and conductor parity. These formulae appear to be new. In particular, we deduce that the first three reduced minimal Weierstrass coefficients of an elliptic curve are determined by its isogeny class.

Problem

• elliptic curves
• Frobenius traces
• Weierstrass coefficients
• decision trees
• isogeny class

Innovation

• Decision trees
• Frobenius traces
• Weierstrass coefficients
• Elliptic curves
• Isogeny class