Torsion and Positive Rank in an Elliptic Family Arising from Cubic 2-Cycles

Abstract

We study the elliptic family $E_t:Y^2=X^3+4X^2+16t^2$ arising from rational $2$-cycles of $x^3+bx+a$. For every $t\in\mathbb{Q}^\times$, we prove that $P_t=(0,-4t)$ has infinite order, so every nonsingular rational fiber has positive rank. We determine its rational torsion subgroup: it is cyclic of order $2$ precisely when $t=u(u^2+4)/4$ for some $u\in\mathbb{Q}^\times$, and is trivial otherwise. No such fiber admits a rational $3$- or $5$-isogeny. An explicit birational dictionary then shows that, for every fixed $a\in\mathbb{Q}^\times$, infinitely many $b\in\mathbb{Q}$ yield a rational $2$-cycle of $x^3+bx+a$. The uniform non-torsion assertion follows from Nagell--Lutz integrality. The torsion exclusions combine elementary $2$-descent with explicit genus-$3$ curves, an unconditional rank-zero Prym argument, and two-cover descent with elliptic Chabauty. Exact Magma and SageMath certificates accompany the computer-assisted steps.