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 $E_t:Y^2=X\left(X^2+\left(\frac{t^2}{4}-2\right)X+1\right).$ . This equation arises from the study of the Mahler measure of polynomials. We show that the rank of $E_t\left(\overline{Q}\right(t\left)\right)$ is 0 and the torsion subgroup of $E_t\left(Q\right(t\left)\right)$ is isomorphic to $Z∕4Z$ . Over the rational field $Q$ 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 $E_t\left(Q\right)$ and conclude with some results regarding integral points in arithmetic progression on E_t .