Glasnik matematički, Vol. 47 No. 1, 2012.
Original scientific paper
https://doi.org/10.3336/gm.47.1.06
On the family of elliptic curves Y^2=X^3-T^2X+1
Petra Tadic
; Martićeva 23, 10000 Zagreb, Croatia
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.
Keywords
Elliptic surface; elliptic curve; parametrization; function field; rank; family of elliptic curves; torsion
Hrčak ID:
82572
URI
Publication date:
3.6.2012.
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