- Hacettepe Journal of Mathematics and Statistics
- Vol: 50 Issue: 3
- Elliptic curve involving subfamilies of rank at least 5 over $\mathbb{Q}(t)$ or $\mathbb{Q}(t,k)$
Elliptic curve involving subfamilies of rank at least 5 over $\mathbb{Q}(t)$ or $\mathbb{Q}(t,k)$
Authors : Ahmed El Amine Youmbai, A. Muhammed Uludağ, Djilali Behloul
Pages : 721-731
Doi:10.15672/hujms.708945
View : 11 | Download : 4
Publication Date : 2021-06-07
Article Type : Research
Abstract :Motivated by the work of Zargar and Zamani, we introduce a family of elliptic curves containing several one- (respectively two-) parameter subfamilies of high rank over the function field $\mathbb{Q}(t)$ (respectively $\mathbb{Q}(t,k)$). Following the approach of Moody, we construct two subfamilies of infinitely many elliptic curves of rank at least 5 over $\mathbb{Q}(t,k)$. Secondly, we deduce two other subfamilies of this family, induced by the edges of a rational cuboid containing five independent $\mathbb{Q}(t)$-rational points. Finally, we give a new subfamily induced by Diophantine triples with rank at least 5 over $\mathbb{Q}(t)$. By specialization, we obtain some specific examples of elliptic curves over $\mathbb{Q}$ with a high rank (8, 9, 10 and 11).Keywords : Elliptic Curves, Rank, Rational Cuboid, Diophantine Triples