Creation of Genus One Models#

GenusOneModel(seq): [ RngElt ] -> ModelG1#

GenusOneModel(n, seq): RngIntElt, [ RngElt ] -> ModelG1#

GenusOneModel(R, n, seq): RngIntElt, [ RngElt ] -> ModelG1#

GenusOneModel(n, str): RngIntElt, MonStgElt -> ModelG1#

The genus one model of degree \(n\) (where \(n\) is 2, 3, 4, or 5) determined by the coefficients in the given sequence or string. The coefficients may belong to any ring.

A sequence \([a,b,c,d,e]\) of length 5 is interpreted as the degree 2 model \(ax^4+bx^3z+cx^2z^2+dxz^3+ez^4\). A sequence \([f,g,h,a,b,c,d,e]\) of length 8 is interpreted as the degree 2 model \(y^2+y(fx^2+gxz+hz^2)-(ax^4+bx^3z+cx^2z^2+dxz^3+ez^4)\).

A sequence \([a,b,c,d,e,f,g,h,i,j]\) of length 10 is interpreted as the degree 3 model \(ax^3+by^3+cz^3+dx^2y+ex^2z+fy^2x+gy^2z+hz^2x+iz^2y+jxyz\).

Sequences of lengths 20 or 50 are interpreted as models of degree 4 or 5 respectively; however, it is easier to create these by specifying matrices instead (see below).

The sequence of coefficients can be recovered by calling Eltseq.

GenusOneModel(C): Crv -> ModelG1#

A genus one model that represents the given curve \(C\).

For degree 2, \(C\) should either be a subscheme of a weighted projective space \({\mathbb{P}}(1,1,2)\), or a hyperelliptic curve. For degrees \(n=3\), 4, or 5, \(C\) should be a genus one normal curve of degree \(n\); in other words, a plane cubic for \(n=3\), an intersection of two quadrics in \({\mathbb{P}}^3\) for \(n=4\), or an intersection of five quadrics in \({\mathbb{P}}^4\) for \(n=5\).

GenusOneModel(f): RngMPolElt -> ModelG1#
GenusOneModel(f): RngUPolElt -> ModelG1#
GenusOneModel(seq): [ RngMPolElt ] -> ModelG1#: The genus one model given by the polynomial \(f\) or the sequence of equations seq.

GenusOneModel(n, E): RngIntElt, CrvEll -> ModelG1, Crv, MapSch, MapSch#: A genus one model of degree \(n\) (where \(n\) is 2, 3, 4, or 5) representing the elliptic curve \(E\) embedded in \({\mathbb{P}}^{n-1}\) via the linear system \(|n.O|\). Also returned are the image of the embedding as a curve \(C\) together with the maps of schemes \(E \to C\) and \(C \to E\).

GenusOneModel(mat): Mtrx -> ModelG1#: The genus one model of degree 5 associated to the given \(5 \times 5\) matrix.

GenusOneModel(mats): [ AlgMatElt ] -> ModelG1#: The genus one model of degree 4 determined by the given pair of \(4 \times 4\) symmetric matrices in the sequence mats. (The matrices can be recovered by calling ModelToMatrices).

IsGenusOneModel(f): RngUPolElt -> BoolElt, ModelG1#

IsGenusOneModel(f): RngMPolElt -> BoolElt, ModelG1#

IsGenusOneModel(seq): [ RngMPolElt ] -> BoolElt, ModelG1#

IsGenusOneModel(mat): Mtrx -> BoolElt, ModelG1#

Returns true if and only if the given polynomial, sequence of polynomials, or matrix determines a “genus one model” in the sense described in the introduction to this chapter. When true, the model is also returned.

Important note: This does not imply that the associated scheme is a curve of genus \(1\). Degenerate models are allowed.

GenericModel(n): RngIntElt -> ModelG1#: The generic genus one model of degree \(n\), where \(n\) is 2, 3, 4 or 5. The coefficients are indeterminates in a suitable polynomial ring.

RandomGenusOneModel(n): RngIntElt -> ModelG1#

RandomModel(n): RngIntElt -> ModelG1#

Size: RngIntElt                    Default: 

A random genus one model of degree \(n\), where \(n\) is 2, 3, 4, or 5. The optional parameter Size is passed to RandomSL or RandomGL.

ChangeRing(model, R): ModelG1, Rng -> ModelG1#: The genus one model defined over the ring \(R\) obtained by coercing the coefficients of the given genus one model into \(R\).

CompleteTheSquare(model): ModelG1 -> ModelG1#: Given a genus one model of degree 2, returns a simplified genus one model of degree 2 without cross terms; this is computed by completing the square on the multivariate polynomial defining the original model.

CubicFromPoint(E, P): CrvEll, PtEll -> RngMPolElt, MapSch, Pt#: The \(3\)-covering corresponding to the rational point \(P\) on an elliptic curve \(E\). The \(3\)-covering is returned as the equation of a projective plane cubic curve. Also returned are the covering map and a point that maps to \(P\) under the covering map.

HesseModel(n, seq): RngIntElt, [ RngElt ] -> ModelG1#: A genus one model of degree \(n\) invariant under the standard representation of the Heisenberg group. The second argument should be a sequence of two ring elements.

DiagonalModel(n, seq): RngIntElt, [ RngElt ] -> ModelG1#: A genus one model of degree \(n\) invariant under the diagonal action of \(\mu_n\). The second argument should be a sequence of \(n\) ring elements.

Example: Generic Model (ex-bc04b8)#

We construct the genus one model of degree 5 obtained from the generic elliptic curve \(E_{a,b} : y^2 = x^3+ax+b\) over \({\mathbb{Q}}(a,b)\). The model is the image of \(E_{a,b}\) under the embedding in \({\mathbb{P}}^4\) given by the linear system \(|5.O|\).

> K<a,b> := FunctionField(Rationals(), 2);
> Eab := EllipticCurve([a, b]);
> model := GenusOneModel(5, Eab);
> model;
[           0 -b*x1 - a*x2           x5           x4           x3]
[ b*x1 + a*x2            0           x4           x3           x2]
[         -x5          -x4            0          -x2            0]
[         -x4          -x3           x2            0           x1]
[         -x3          -x2            0          -x1            0]

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From this matrix, which is the data storing the model, the equations of the curve in \({\mathbb{P}}^4\) can be computed; they are quadratic forms given by the \(4 \times 4\) Pfaffians of the matrix.

> Equations(model);
[
    -x1*x4 + x2^2,
    x1*x5 - x2*x3,
    b*x1^2 + a*x1*x2 + x2*x4 - x3^2,
    -x2*x5 + x3*x4,
    -b*x1*x2 - a*x2^2 + x3*x5 - x4^2
]

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Note that the degree \(5\) model has the same invariants \(c4, c6, \Delta\) as \(E_{a,b}\):

> Invariants(model);
-48*a
-864*b
-64*a^3 - 432*b^2
> cInvariants(Eab), Discriminant(Eab);
[
    -48*a,
    -864*b
]
-64*a^3 - 432*b^2

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