^{}aDepartment of mathematics, Arak Branch, Islamic Azad university, Arak, Iran.

Abstract

In this paper the family of elliptic curves over Q given by the equation Ep : Y^{2} = (X - p)^{3} + X^{3} + (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.

Article Title [Persian]

On the rank of certain parametrized
elliptic curves

Authors [Persian]

Ali Hadavand

^{}aDepartment of mathematics, Arak Branch, Islamic Azad university, Arak,
Iran.

Abstract [Persian]

In this paper the family of elliptic curves over Q given by the equation Ep : Y^{2} = (X - p)^{3} + X^{3} + (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.

Keywords [Persian]

Elliptic Curve، Selmer Group

References

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