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Rad Hrvatske akademije znanosti i umjetnosti. Matematičke znanosti, No. 555=27, 2023.

Original scientific paper

https://doi.org/10.21857/ydkx2coje9

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

Lorenz Halbeisen ; Department of Mathematics, ETH Zentrum Rämistrasse 101, 8092 Zürich, Switzerland
Norbert Hungerbühler orcid id orcid.org/0000-0001-6191-0022 ; Department of Mathematics, ETH Zentrum Rämistrasse 101, 8092 Zürich, Switzerland
Arman Shamsi Zargar ; Department of Mathematics and Applications, University of Mohaghegh Ardabili, Ardabil, Iran
Maksym Voznyy orcid id orcid.org/0000-0001-5649-8416 ; Department of Technology, Stephen Leacock CI, Toronto District School Board, Toronto, Canada

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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.

Keywords

Elliptic curve; parametrisation; quadratic field; rank; torsion group

Hrčak ID:

307488

URI

https://hrcak.srce.hr/307488

Publication date:

25.8.2023.

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