Families of elliptic curves over cyclic cubic number fields with prescribed torsion (Q2792373)

The main goal in this article is to construct infinite families of elliptic curves having particular groups as torsion groups over cyclic cubic fields. These particular groups came up in the author's et al. previous work [Math. Comput. 80, No. 273, 579--591 (2011; Zbl 1214.11071)] and joint work with \textit{C. H. Kim} and \textit{A. Schweizer} [Acta Arith. 113, No. 3, 291--301 (2004; Zbl 1083.11038)], where they proved that only these groups occur infinitely often as torsion groups of elliptic curves over cubic number fields. The method of construction in the article under review is based on realizing suitable modular curves as degree \(3\) cyclic covers of \(\mathbb{P}^{1}\).