Jacobians

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 \({\mathbb{Q}}\).

Creation of a Jacobian

Jacobian(C): CrvHyp -> JacHyp

The Jacobian of the hyperelliptic curve \(C\).

Access Operations

Curve(J): JacHyp -> CrvHyp

The hyperelliptic curve from which the Jacobian \(J\) was constructed.

Dimension(J): JacHyp -> RngIntElt

The dimension of the Jacobian \(J\) as an algebraic variety, equal to the genus of the curve \(C\) of which \(J\) is the Jacobian.

Base Ring

BaseField(J): JacHyp -> Fld

BaseRing(J): JacHyp -> Rng

CoefficientRing(J): JacHyp -> Rng

The base field of the Jacobian \(J\).

Changing the Base Ring

BaseChange(J, F): JacHyp, Rng -> JacHyp

BaseExtend(J, F): JacHyp, Rng -> JacHyp

The base extension of the Jacobian \(J\) to the field \(F\).

BaseChange(J, j): JacHyp, Map -> JacHyp

BaseExtend(J, j): JacHyp, Map -> JacHyp

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.

BaseChange(J, n): JacHyp, RngIntElt -> JacHyp

BaseExtend(J, n): JacHyp, RngIntElt -> JacHyp

The base extension of the Jacobian \(J\) over a finite field to its degree \(n\) extension.