Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves
2026-07-27 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study how to find certain important numbers called reduced minimal Weierstrass coefficients that define an elliptic curve, using data from something called Frobenius traces. They use decision trees to show that the first two coefficients can be exactly figured out from Frobenius traces at the primes 2 and 3, and the third coefficient can be found if they also know whether the curve’s conductor is odd or even. They then prove new formulae that link these coefficients to the Frobenius traces and conductor parity. Their work shows these coefficients depend only on the curve’s isogeny class, a way of grouping elliptic curves.
Elliptic curveReduced minimal Weierstrass coefficientsFrobenius tracePrime numbersConductor parityIsogeny classDecision treeNumber theoryMinimal model
Authors
Barinder S. Banwait, Xiaoyu Huang, Kyu-Hwan Lee, Seewoo Lee, Thomas Oliver, Alexey Pozdnyakov
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.