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...
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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
Et(
\(\overline{\mathbb{Q}}\)(
t)) is 0 and the torsion subgroup of
Et(
\(\mathbb{Q}\)(
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
Et with rank 5 and 6. We also determine all possible torsion subgroups of
Et(
\(\mathbb{Q}\)) and conclude with some results regarding integral points in arithmetic progression on
Et.
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