A superelliptic curve is a generalization of an elliptic curve defined by an equation of the form: \[ y^m = P(x) \] where \( P(x) \) is a polynomial in \( x \) of degree \( n \), and \( m \) is a positive integer typically greater than 1.
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A superelliptic curve has an affine equation . When is square-free and the characteristic does not divide , projection to the -coordinate is a finite morphism whose finite ramification occurs over the roots of ; its behavior at infinity depends on and .