On the Elliptic Curves of the Form y2=x3−pqxy^2 = x^3 − pqxy2=x3−pqx (original) (raw)
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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.
Elliptic Curves of Type y2=x3−3pqx Having Ranks Zero and One
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The group of rational points on an elliptic curve over Q is always a finitely generated Abelian group, hence isomorphic to Zr×G with G a finite Abelian group. Here, r is the rank of the elliptic curve. In this paper, we determine sufficient conditions that need to be set on the prime numbers p and q so that the elliptic curve E:y2=x3−3pqx over Q would possess a rank zero or one. Specifically, we verify that if distinct primes p and q satisfy the congruence p≡q≡5(mod24), then E has rank zero. Furthermore, if p≡5(mod12) is considered instead of a modulus of 24, then E has rank zero or one. Lastly, for primes of the form p=24k+17 and q=24ℓ+5, where 9k+3ℓ+7 is a perfect square, we show that E has rank one.
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The Mordell-Weil theorem states that the group of rational points on an elliptic curve over the rational numbers is a finitely generated abelian group. In our previous paper, H. Daghigh, and S. Didari, On the elliptic curves of the form $ y^2=x^3-3px$, Bull. Iranian Math. Soc. 40 (2014), no. 5, 1119--1133., using Selmer groups, we have shown that for a prime ppp the rank of elliptic curve y2=x3−3pxy^2=x^3-3pxy2=x3−3px is at most two. In this paper we go further, and using height function, we will determine the Mordell-Weil group of a family of elliptic curves of the form y2=x3−3nxy^2=x^3-3nxy2=x3−3nx, and give a set of its generators under certain conditions. We will introduce an infinite family of elliptic curves with rank at least two. The full Mordell-Weil group and the generators of a family (which is expected to be infinite under the assumption of a standard conjecture) of elliptic curves with exact rank two will be described.
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