Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves
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Abstract (arXiv)
We investigate the extent to which the coefficients (w1,w2,w3,w4,w6) of the reduced minimal Weierstrass model of an elliptic curve E/mathbbQ 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.
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References · 10
- 2020Machine-Learning the Sato--Tate Conjecturecited by 14
- 2022Murmurations of elliptic curvescited by 14
- 2020Machine-Learning Arithmetic Curvescited by 10
- 2019Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyercited by 8
- 2022Machine Learning Class Numbers of Real Quadratic Fieldscited by 6
- 2025Learning Euler Factors of Elliptic Curvescited by 4
- 2024Machine Learning Approaches to the Shafarevich-Tate Group of Elliptic Curvescited by 2
- 2025Machines Learn Number Fields, But How? The Case of Galois Groupscited by 2
- 2026Murmurations, Mestre--Nagao sums, and Convolutional Neural Networks for elliptic curvescited by 2
- 2025Learning Fricke signs from Maass form Coefficientscited by 1
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