ON THE FAMILY OF ELLIPTIC CURVES X + 1/X + Y + 1/Y + t = 0
Dustin Moody, Abhishek Juyal
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.
INTEGERS, The electronic journal of combinatorial number theory
and Juyal, A.
ON THE FAMILY OF ELLIPTIC CURVES X + 1/X + Y + 1/Y + t = 0, INTEGERS, The electronic journal of combinatorial number theory, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=931852
(Accessed December 2, 2022)