ScienceAsia 40S (2014): 69-72 |doi:
Towards the construction of elliptic curves over ℚ with high rank and a point of order 5
Fahd M. Alshammari
ABSTRACT: By representing a genus one curve as a plane curve with five double points, we are able to construct a 3-parameter family of genus one curves over ℚ with Jacobians having a torsion subgroup isomorphic to Z5. This leads, by specializing the parameters, to elliptic curves over ℚ of the Mordell-Weil group with high rank and with a torsion subgroup isomorphic to mathbbZ5. We also show this family contains as a subfamily the principal homogeneous space parameterizing elliptic curves with a rational point of order 5, namely X1(5). We explicitly describe these families by equations in the Weierstrass form.
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|King Suad University, College of Science, Department of Mathematics, P.O.Box 2455 Riyadh 11451, Saudi Arabia
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Received 13 Feb 2014, Accepted 0 0000