Rational 2-Cycles for $x^3+bx+a$ and the Elliptic Family $Y^2=X^3+4X^2+16t^2$

Abstract

We study rational $2$-cycles for the cubic family $f_{b,a}(x)=x^3+bx+a$, where $a,b\in\mathbb{Q}$, via the arithmetic of elliptic curves. For fixed $a\neq0$, we give explicit birational formulas relating rational $2$-cycles of $f_{b,a}$ to rational points on $E_a:Y^2=X^3+4X^2+16a^2$. This specializes the known normal-form-II elliptic period-$2$ locus to the square-coefficient slice arising from $f_{b,a}$. Using explicit division-polynomial evaluations and Mazur's theorem, we prove that the distinguished section $P_t=(0,-4t)$ has infinite order on $\mathcal{E}_t:Y^2=X^3+4X^2+16t^2$ for every $t\in\mathbb{Q}^{\times}$. Consequently $E_a(\mathbb{Q})$ has positive rank for every nonzero $a\in\mathbb{Q}$, and for each such $a$ there are infinitely many rational values of $b$ for which $f_{b,a}$ admits a rational $2$-cycle. We then undertake a finer arithmetic study of the generic family $\mathcal{E}_t$ over $\mathbb{Q}$: we determine its rational $2$-torsion locus and prove that for $t\neq0$ no specialization admits a rational $5$-isogeny or a rational point of order $3$, $5$, or $7$; the order-$5$ and order-$7$ exclusions rest on certified Magma computations on explicit genus-$3$ curves. The torsion analysis is logically independent of the infinite-order theorem and is not required for the dynamical application.