On the Family of Elliptic Curves X + 1/X + Y + 1/Y + t = 0

We study various properties of the family of elliptic curves \(x + 1/x + y + 1/y + t = 0\), which is isomorphic to the Weierstrass curve \(E_{t} : Y^2 = X\left(X^2+\middle(\frac{t^2}{4}-2\right) X+1)\). This equation arises from the study of the Mahler measure of polynomials. We show that the rank... See full abstract

We study various properties of the family of elliptic curves \(x + 1/x + y + 1/y + t = 0\), which is isomorphic to the Weierstrass curve \(E_{t} : Y^2 = X\left(X^2+\middle(\frac{t^2}{4}-2\right) X+1)\). This equation arises from the study of the Mahler measure of polynomials. We show that the rank of E_t(\(\overline{\mathbb{Q}}\)(t)) is 0 and the torsion subgroup of E_t(\(\mathbb{Q}\)(t)) is isomorphic to \(\mathbb{Z}\)/4\(\mathbb{Z}\). Over the rational field \(\mathbb{Q}\) we obtain infinite subfamilies of ranks (at least) one and two, and find specific instances of E_t with rank 5 and 6. We also determine all possible torsion subgroups of E_t(\(\mathbb{Q}\)) and conclude with some results regarding integral points in arithmetic progression on E_t.
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