Elliptic curves are objects of algebraic geometry met in somewhat advanced parts of number theory. They also appear in applications to cryptography. Use the tag, if this applies. Questions on ellipses should be tagged [conic-sections] instead.

An elliptic curve is a smooth, nonsingular projective curve of genus 1 with a specified point $\mathcal{O}$. It forms an abelian group under point addition. It is an object of much study in number theory, for example in cryptography and integer factorization.

More informally, elliptic curves are cubic curves that have a shape that looks like a donut. They can be defined over any field $K$.

An elliptic curve can be defined by an equation of the form: $$E:y^2=x^3+ax+b$$ with the discriminant $\triangle_E=16(4a^3+27b^2)\ne 0$ so the curve is nonsingular, i.e. its graph has no cusps or intersections.

The elliptic curves with $a=0$ are mordell-curves.