On the rank of certain parametrized elliptic curves

Document Type: Research Articles


aDepartment of mathematics, Arak Branch, Islamic Azad university, Arak, Iran.


In this paper the family of elliptic curves over Q given by the equation Ep :
Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. It
is shown that the maximal rank of the elliptic curves is at most 3 and some
conditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 or
rank(Ep(Q))≥2 are given.

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