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.
