Hyperelliptic Curve

A hyperelliptic curve is an algebraic curve given by an equation of the form , where  is a polynomial of degree  with  distinct roots.

If  is a cubic or quartic polynomial, then the curve is called an elliptic curve.

The genus of a hyperelliptic curve is related to the degree of the polynomial. A polynomial of degree  or  gives a curve of genus . All curves of genus 2 are hyperelliptic, but for larger genus there are curves that are not hyperelliptic.