Editing Hyperelliptic curve (section)

== Using Riemann–Hurwitz formula ==

Using the [[Riemann–Hurwitz formula]], the hyperelliptic curve with genus ''g'' is defined by an equation with degree ''n'' = 2''g'' + 2. Suppose  ''f'' : ''X'' → P<sup>1</sup> is a branched covering with ramification degree ''2'', where ''X'' is a curve with genus ''g'' and P<sup>1</sup> is the [[Riemann sphere]]. Let ''g''<sub>1</sub> = ''g'' and ''g''<sub>0</sub> be the genus of P<sup>1</sup> ( = 0 ), then the Riemann-Hurwitz formula turns out to be
:<math>2-2g_1 =2(2-2g_0)-\sum_{s \in X}(e_s-1)</math>
where ''s'' is over all ramified points on ''X''. The number of ramified points is ''n'', and at each ramified point ''s'' we have ''e<sub>s</sub>'' = 2, so the formula becomes
:<math>2-2\times g =2(2-2\times0)-n\times(2-1)</math>
so ''n'' = 2''g'' + 2.