Families of Elliptic Curves and Geometric Properties of Torsion Points
In this chapter we consider families of elliptic curves by studying cubics in normal form with coefficients depending on a parameter. The most important example is the Legendre family E: y 2 = x(x-1)(x-λ) over k(λ), where k is a field of characteristic unequal to 2. The points ·(0, 0), (1, 0), (λ, 0)× are the three 2-torsion points on E λ for each value of λ ∈ k - ·0, 1×, and they are specified with a given ordering.
KeywordsNormal Form Elliptic Curve Elliptic Curf Tangent Line Double Point
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