Generators and integral points on elliptic curves associated with simplest quartic fields

Abstract

Abstract We associate to some simplest quartic fields a family of elliptic curves that has rank at least three over ℚ(m). It is given by the equation E m : y 2 = x 3 − 36 36 m 4 + 48 m 2 + 25 36 m 4 − 48 m 2 + 25 x . $$\begin{array}{} \displaystyle E_m:y^2=x^3-36\left(36m^4+48m^2+25\right)\left(36m^4-48m^2+25\right)x. \end{array}$$ Employing canonical heights we show the rank is in fact at least three for all m. Moreover, we get a parametrized infinite family of rank at least four. Further, the integral points on the curve Em are discussed and we determine all the integral points on the original quartic model when the rank is three. Previous work in this setting studied the elliptic curves associated with simplest quartic fields of ranks at most two along with their integral points (see [2, 3]).