Elliptic curves with torsion groups $\mathbb{Z}/8\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/6\mathbb{Z}$

Abstract

In this paper, we present details of seven elliptic curves over $\mathbb{Q}(u)$ with rank $2$ and torsion group $\mathbb{Z}/8\mathbb{Z}$ and five curves over $\mathbb{Q}(u)$ with rank $2$ and torsion group $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/6\mathbb{Z}$. We also exhibit some particular examples of curves with high rank over $\mathbb{Q}$ by specialization of the parameter. We present several sets of infinitely many elliptic curves in both torsion groups and rank at least $3$ parametrized by elliptic curves having positive rank. In some of these sets we have performed calculations about the distribution of the root number. This has relation with recent heuristics concerning the rank bound for elliptic curves by Park, Poonen, Voight and Wood.

arXiv comment22 pages, 3 figures; revised according to referees remarks; extended figures in Section 11, based on the factorizations obtained by the members of Mersenne Forum; to appear in RACSAM

References in corpus (5)

• On Mordell-Weil groups of elliptic curves induced by Diophantine triples
• Elliptic curves induced by Diophantine triples
• Elliptic curves with torsion groups $\mathbb{Z}/8\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/6\mathbb{Z}$
• On the torsion group of elliptic curves induced by Diophantine triples over quadratic fields
• A Simple Method for High-Rank Families of Elliptic Curves with Specified Torsion

Cited by in corpus (1)

• Elliptic curves with torsion groups $\mathbb{Z}/8\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/6\mathbb{Z}$