ON THE FAMILY OF ELLIPTIC CURVES X + 1/X + Y + 1/Y + t = 0

Published

March 23, 2021

Author(s)

Dustin Moody, Abhishek Juyal

Abstract

We study various properties of the family of elliptic curves x+ 1/x+y+ 1/y+t = 0, which is isomorphic to the Weierstrass curve E_t: Y^2=X(X^2+(t^2/4-2)X+1). This equation arises from the study of the Mahler measure of polynomials. We show that the rank of Et(Q(t)) is 0 and the torsion subgroup of Et(Q(t)) is isomorphic to Z/4Z. Over the rational field Q we obtain infinite subfamilies of ranks (at least) one and two, and find specific instances of Et with rank 5 and 6. We also determine all possible torsion subgroups of Et(Q) and conclude with some results regarding integral points in arithmetic progression on Et.

Citation

INTEGERS, The electronic journal of combinatorial number theory

Volume

21

Pub Type

Journals

Download Paper

https://doi.org/10.5281/zenodo.10807511

Local Download

Keywords

Elliptic Curves, Mahler measure, Rank

Citation

Moody, D. and Juyal, A. (2021), ON THE FAMILY OF ELLIPTIC CURVES X + 1/X + Y + 1/Y + t = 0, INTEGERS, The electronic journal of combinatorial number theory, [online], https://doi.org/10.5281/zenodo.10807511, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=931852 (Accessed October 18, 2025)