An elliptic curve is a smooth, projective algebraic curve of genus one, equipped with a specified point, often denoted as O. These curves can be defined over any field and are characterized by their cubic equations of the form $$y^2 = x^3 + ax + b$$, where the discriminant \( \Delta = 4a^3 + 27b^2 \neq 0 \) ensures no singular points exist. Elliptic curves play a crucial role in number theory and algebraic geometry, especially in the study of abelian varieties, where they serve as the simplest examples.
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