Elliptic Curves Arising from Triangular Numbers

Juyal, Abhishek; Kumar, Shiv; Moody, Dustin

Journal Article

Elliptic Curves Arising from Triangular Numbers

Published: February 1, 2019
Citation: INTEGERS: The electronic journal of combinatorial number theory vol. 19, article no. A10 (February 2019) pp. 1-11

Author(s)

Abhishek Juyal (Harish-Chandra Research Institute), Shiv Kumar (Motilal Nehru National Institute of Technology), Dustin Moody (NIST)

Abstract

We study the Legendre family of elliptic curves E_t : y² = 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.

We study the Legendre family of elliptic curves Et : y2 = x(x − 1)(x − ∆t), parametrized by triangular numbers ∆t = t(t + 1)/2. We prove that the rank of Et 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... See full abstract

Keywords

elliptic curves; triangular numbers

Control Families

None selected

Documentation

Publication:
Volume 19 (w/ link to paper)

Supplemental Material:
Preprint (pdf)

Document History:
02/01/19: Journal Article (Final)

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