Imaginary hyperelliptic curve

A hyperelliptic curve is a particular kind of algebraic curve. There exist hyperelliptic curves of every genus ${\displaystyle g\geq 1}$. If the genus of a hyperelliptic curve equals 1, we simply call the curve an elliptic curve. Hence we can see hyperelliptic curves as generalizations of elliptic curves. There is a well-known group structure on the set of points lying on an elliptic curve over some field ${\displaystyle K}$, which we can describe geometrically with chords and tangents. Generalizing this group structure to the hyperelliptic case is not straightforward. We cannot define the same group law on the set of points lying on a hyperelliptic curve, instead a group structure can be defined on the so-called Jacobian of a hyperelliptic curve. The computations differ depending on the number of points at infinity. Imaginary hyperelliptic curves are hyperelliptic curves with exactly 1 point at infinity: real hyperelliptic curves have two points at infinity.