On the elliptic curves of the form $ y^2=x^3-3px $
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Abstract:
By the Mordell-Weil theorem, the group of rational points on an elliptic curve over a number field is a finitely generated abelian group. There is no known algorithm for finding the rank of this group. This paper computes the rank of the family $ E_p:y^2=x^3-3px $ of elliptic curves, where p is a prime.
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on the elliptic curves of the form $ y^2=x^3-3px $
by the mordell-weil theorem, the group of rational points on an elliptic curve over a number field is a finitely generated abelian group. there is no known algorithm for finding the rank of this group. this paper computes the rank of the family $ e_p:y^2=x^3-3px $ of elliptic curves, where p is a prime.
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On the rank of certain parametrized elliptic curves
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. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.
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Journal title
volume 40 issue 5
pages 1119- 1133
publication date 2014-10-01
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