Attributes of Genus One Models

Degree(model): ModelG1 -> RngIntElt

The degree (2, 3, 4, or 5) of the given model.

DefiningEquations(model): ModelG1 -> [ RngMPolElt ]

A sequence containing the equations by which the given genus one model (which must have degree 2, 3, or 4) is defined.

Equations(model): ModelG1 -> [ RngMPolElt ]

A sequence containing equations for the scheme associated to the given genus one model (of any degree). For degree 2, 3, or 4 this is the same as DefiningEquations.

Matrix(model): ModelG1 -> Mtrx

The defining matrix of a genus one model of degree 5.

Curve(model): ModelG1 -> Crv

HyperellipticCurve(model): ModelG1 -> CrvHyp

QuadricIntersection(model): ModelG1 -> Crv

The curve associated to the given genus one model.

In the degree \(2\) case the curve is hyperelliptic of the form \(y^2 + f(x,z) y - g(x,z) = 0\), but is only created explicitly as a hyperelliptic curve when HyperellipticCurve is called; otherwise it is created as a general curve in a weighted projective space in which the variables \(x\), \(z\), and \(y\) have weights 1, 1, and 2).

In the degree 4 case the curve is an intersection of two quadrics in \({\mathbb{P}}^3\).

An error results in degenerate cases where the equations of the model do not define a curve

Matrices(model): ModelG1 -> [ AlgMatElt ]

For a genus one model of degree 4 this function returns a sequence containing two \(4 \times 4\) symmetric matrices representing the quadrics.

BaseRing(model): ModelG1 -> Rng

The coefficient ring of the given model.

PolynomialRing(model): ModelG1 -> RngMPol

The polynomial ring used to define the model.

ModelToSequence(model): ModelG1 -> [ RngElt ]

Eltseq(model): ModelG1 -> [ RngElt ]

A sequence containing the defining coefficients of the given genus one model.

ModelToString(model): ModelG1 -> MonStgElt

A string containing the defining coefficients of the given genus one model.