Let K be a field, which for simplicity we take of characteristic different to 2 and 3. Then an elliptic curve over K is a projective curve defined by an equation of the form ZY2 = X3 + aZ2X + bZ3, where a,b in K are such that 4a3 + 27b2 0.A distinguished feature of elliptic curves is that their points have the structure of an abelian group. This fact makes the theory of elliptic curves extremely interesting and rich, with links to various important notions of mathematics such as modular forms steroid, Galois representations, and sphere packing.

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