Rad HAZU, Matematičke znanosti, Vol. 27 (2023), 87-109.

A GEOMETRIC APPROACH TO ELLIPTIC CURVES WITH TORSION GROUPS Z/10Z, Z/12Z, Z/14Z, and Z/16Z

Lorenz Halbeisen, Norbert Hungerbühler, Arman Shamsi Zargar and Maksym Voznyy

Department of Mathematics, ETH Zentrum Rämistrasse 101, 8092 Zürich, Switzerland
e-mail: norbert.hungerbuehler@math.ethz.ch

Department of Mathematics and Applications, University of Mohaghegh Ardabili, Ardabil, Iran
e-mail: zargar@uma.ac.ir

Department of Technology, Stephen Leacock CI, Toronto District School Board, Toronto, Canada
e-mail: maksym.voznyy@tdsb.on.ca

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.

2020 Mathematics Subject Classification.   11G05, 14H52.

Key words and phrases.   Elliptic curve, parametrisation, quadratic field, rank, torsion group.

DOI: https://doi.org/10.21857/ydkx2coje9

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