We study the Legendre family of elliptic curves E_t : y^2 = x(x − 1)(x − ∆t), parametrized by triangular numbers ∆t = t(t + 1)/2. We prove that the rank of E_t over the function field Q(t) is 1, while the rank is 0 over Q(t). We also produce some infinite subfamilies whose Mordell-Weil rank is positive, and find high rank curves from within these families.
INTEGERS, The electronic journal of combinatorial number theory
, Kumar, S.
and Moody, D.
Elliptic Curves Arising from Triangular Numbers, INTEGERS, The electronic journal of combinatorial number theory, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=927507
(Accessed December 8, 2023)