We implement and interpret various supervised learning experiments involving real quadratic fields with class numbers 1, 2 and 3. We quantify the relative difficulties in separating class numbers of matching/different parity from a data-scientific perspective, apply the methodology of feature analysis and principal component analysis, and use symbolic classification to develop machine-learned formulas for class numbers 1, 2 and 3 that apply to our dataset.

Keywords:

AMSC: 11R29, 11R80, 62R99

References

1. M. Amir (2022), https://github.com/malik-amir/Machine-Learning-of-the-Tate-Shafarevich-group. Google Scholar
2. W. Duke , Extreme values of Artin L-functions and class numbers, Compos. Math. 136(1) (2003) 103–115. Crossref, Google Scholar
3. Y.-H. He, K.-H. Lee and T. Oliver , Machine-learning the Sato–Tate conjecture, J. Symb. Comput. 111 (2022) 61–72. Crossref, Google Scholar
4. Y.-H. He, K.-H. Lee and T. Oliver , Machine-learning number fields, Math. Comput. Geom. Data 2 (2022) 49–66. Crossref, Google Scholar
5. Y.-H. He, K.-H. Lee and T. Oliver , Machine learning the invariants of arithmetic curves, J. Symb. Comput. 115 (2023) 478–491. Crossref, Google Scholar
6. Y.-H. He, K.-H. Lee, T. Oliver and A. Pozdnyakov, Murmurations of elliptic curves, preprint (2022), arXiv:2204.10140. Google Scholar
7. G. J. Janusz , Algebraic Number Fields, Pure and Applied Mathematics, Vol. 55 (Academic Press, New York, 1973). Google Scholar
8. M. F. Korns , Genetic programming symbolic classification: A study, Genetic Programming Theory and Practice XV (Springer, 2018), pp. 39–54. Crossref, Google Scholar
9. E. Kowalski , Elementary theory of L-functions, II, An Introduction to the Langlands Program (Jerusalem, 2001) (Birkhäuser Boston, Boston, MA, 2003), pp. 21–37. Google Scholar
10. F. Lemmermeyer , Reciprocity Laws: From Euler to Eisenstein, Springer Monographs in Mathematics (Springer-Verlag, Berlin, 2000). Crossref, Google Scholar
11. The LMFDB Collaboration, The L-functions and modular forms database (2022), http://www.lmfdb.org. Google Scholar
12. A. Płońska and P. Płoński, MLJAR: State-of-the-art automated machine learning framework for tabular data, Version 0.10.3, MLJAR (2021), https://github.com/mljar/mljar-supervised. Google Scholar
13. S. Udrescu and M. Tegmark , AI Feynman: A physics-inspired method for symbolic regression, Sci. Adv. 6(16) (2020) eaay2631, arXiv:1905.11481. Crossref, Google Scholar
14. S. Wagner et al., Architecture and design of the heuristiclab optimization environment, Advanced Methods and Applications in Computational Intelligence, Topics in Intelligent Engineering and Informatics Series (Springer, 2014), pp. 197–261. Crossref, Google Scholar