A geometric approach to elliptic curves with torsion groups ℤ/10ℤ, ℤ/12ℤ, ℤ/14ℤ, and ℤ/16ℤ

2021/06/12 by Lorenz Halbeısen, Lorenz Halbeisen, Norbert Hungerbuehler +6
Arts and Humanities · Computer Science · Mathematics · Social Sciences · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Historical and Political Studies #North African History and Literature #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.2106.06861

21 pages, 1 figure

openalex publication_date 2021/06/12 · arxiv created 2021/11/18 · arxiv updated 2021/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give new parametrisations of elliptic curves in Weierstrass normal form y²=x³+ax²+bx with torsion groups ℤ/10ℤ and ℤ/12ℤ over ℚ, and with ℤ/14ℤ and ℤ/16ℤ over quadratic fields. Even though the parametrisations are equivalent to those given by Kubert and Rabarison, respectively, with the new parametrisations we found three infinite families of elliptic curves with torsion group ℤ/12ℤ and positive rank. Furthermore, we found elliptic curves with torsion group ℤ/14ℤ and rank 3, which is a new record for such curves, as well as some new elliptic curves with torsion group ℤ/16ℤ and rank 3.

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