On the Equation

We consider certain quartic twists of an elliptic curve. We establish the rank of these curves under the Birch and Swinnerton-Dyer conjecture and obtain bounds on the size of Shafarevich-Tate group of these curves. We also establish a reduction between the problem of factoring integers of a certain form and the problem of computing rational points on these twists. 1. Introduction In this paper, we investigate certain quartic twists of the elliptic curve and present some of their interesting properties. Specifically, we consider the family of elliptic curves , where with and distinct prime numbers, . These elliptic curves have complex multiplication by . The -torsion point generates the torsion subgroup of the Mordell-Weil group . Our first result concerns the rank of and an interesting valuational property of points in . More specifically we obtain the following. Theorem 1. Let , where with and distinct prime numbers and . Then the rank of the Mordell-Weil group is less than or equal to 1. If the rank is one, then for every point of which is not in , the -adic and -adic valuations of must have opposite parity. Moreover, under the Birch and Swinnerton-Dyer conjecture, the rank of is one. The situation where and do not satisfy the congruence condition in Theorem 1 is less clear. Recently Li and Zeng [1] showed, under the Birch and Swinnerton-Dyer conjecture, that, for where and are distinct odd primes, there exists an elliptic curve , where depends on the classes of and modulo 8, such that the elliptic curve has rank 1 and the valuations at and of -coordinate are not equal for odd , where is a generator of the Mordell-Weil group . By assuming conjectures, in addition to the Birch and Swinnerton-Dyer conjecture, we also obtain the following. Theorem 2. Let , where with and distinct prime numbers and . Then the following holds under the Birch and Swinnerton-Dyer conjecture, the elliptic curve analog of the Brauer-Siegel theorem, and the Hardy-Littlewood’s F conjecture. For every there are infinitely many with and and prime with , such that , where is the Shafarevich-Tate group of and is the minimal discriminant of . Let be an elliptic curve over . The na？ve height of the elliptic curve is defined to be where and are the -invariants associated to a minimal model of . Let vary over a family of number fields of a fixed extension degree over . Let , , and denote, respectively, the discriminant, the class number, and the regulator of . The Brauer-Siegel theorem [2] states that if tends to infinity then . The elliptic curve analog of the Brauer-Siegel Theorem [3]