The Jacobian of a hyperelliptic curve is implemented as the divisor class group of the curve. In particular, no equations giving the Jacobian as a variety ever appear. The Jacobian of any hyperelliptic curve can be created, but most of the interesting functionality is over finite fields, or for genus 2 over number fields or Q.
The Jacobian of the hyperelliptic curve C.
The hyperelliptic curve from which the Jacobian J was constructed.
The dimension of the Jacobian J as an algebraic variety, equal to the genus of the curve C of which J is the Jacobian.
The base field of the Jacobian J.
The base extension of the Jacobian J to the field F.
The base extension of the Jacobian J obtained by the map j, where j is a ring homomorphism with the base field of C as its domain.
The base extension of the Jacobian J over a finite field to its degree n extension.