On the Equation $Y^2=(X+p)(X^2+p^2)$

Abstract

In this paper the family of elliptic curves over Q given by the equation y 2 = (x + p)(x 2 + p 2 ) is studied. It is shown that for p a prime number 3 mod 8, the only rational solution to the equation given here is the one with y = 0. The same is true for p = 2. Standard conjectures predict that the rank of the group of rational points is odd for all other primes p. A lot of numerical evidence in support of this is given. We show that the rank is bounded by 3 in general for prime numbers p. Moreover, this bound can only be attained for certain special prime numbers p 1 mod 16. Examples of such rank 3 curves are given. Lastly, for certain primes p 9 mod 16 nontrivial elements in the Shafarevich group of the elliptic curve are constructed. In the literature one finds similar investigations of elliptic curves with complex multiplication. It may be interesting to note that the curves considered here do not admit complex multiplication.