Glasnik Matematicki, Vol. 47, No. 1 (2012), 81-93.

ON THE FAMILY OF ELLIPTIC CURVES Y2=X3-T2X+1

Petra Tadić

Martićeva 23, 10000 Zagreb, Croatia
e-mail: petra.tadic.zg@gmail.com

Abstract.   Let E be the elliptic curve over Q(T) given by the equation

E:Y2=X3-T2X+1.
We prove that the torsion subgroup of the group E(C(T)) is trivial, rankQ(T)(E)=3 and rankC(T)(E)=4. We find a parametrization of E of rank at least four over the function field Q(a,i,s,n,k) where s2=i3-a2i. From this we get a family of rank ≥ 5 over the field of rational functions in two variables and a family of rank ≥ 6 over an elliptic curve of positive rank. We also found particular elliptic curves with rank ≥ 11.

2010 Mathematics Subject Classification.   11G05, 14H52.

Key words and phrases.   Elliptic surface, elliptic curve, parametrization, function field, rank, family of elliptic curves, torsion.

Full text (PDF) (free access)

DOI: 10.3336/gm.47.1.06

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