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Research articles

ScienceAsia 40S (2014): 69-72 |doi: 10.2306/scienceasia1513-1874.2014.40S.069

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

* Corresponding author, E-mail: fmobarak@ksu.edu.sa

Received 13 Feb 2014, Accepted 0 0000