Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$ via a Rank-Zero Elliptic Curve

Abstract

In this paper we study the even monic degree-8 cuboid polynomial $P_{a,u}(t)$ introduced by R.A. Sharipov in the first-cuboid specialization of his cuboid equations. For nonzero integers $a,u$ with $u^2\ne a^2$ we prove that $P_{a,u}(t)$ is irreducible in $\mathbb{Z}[t]$ (equivalently, in $\mathbb{Q}[t]$), thus confirming Sharipov's irreducibility conjecture in this two-parameter case. Over $K=\mathbb{Q}(\sqrt{2})$ we have a factorization $P_{a,u}(t)=H_{-}(t)H_{+}(t)$ into two conjugate quartics. We show that any further factorization of $H_{\pm }$ would force the discriminant of a certain quadratic in $S=t^2$ to be a square in $K$, which in turn implies (via $\tau =(au/\Delta {)}^2$) the existence of a rational point $(y,v)\in \mathcal{C}(\mathbb{Q})$ on the genus-one quartic $\mathcal{C}:v^2=16y^4+136y^2+1$ with $y^2=\tau$. We give an explicit isomorphism $\overline{\mathcal{C}}\simeq E$ with the elliptic curve $E:Y^2=X(X-8)(X-9)$, whose Mordell-Weil group has rank $0$ and conductor $48$. Enumerating $E(\mathbb{Q})$ and tracing back to $\mathcal{C}(\mathbb{Q})$ rules out the only possible values $\tau \in \{0,\frac{1}{4}\}$, and hence excludes any factorization in $K[t]$. A quadratic Galois descent then yields the irreducibility of $P_{a,u}(t)$ over $\mathbb{Q}$ and $\mathbb{Z}$.

A complete short proof of the first perfect cuboid conjecture (Sharipov's first conjecture)