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 ax4 + bx3z + cx2z2 + dxz3 + ez4. A sequence [f, g, h, a, b, c, d, e] of length 8 is interpreted as the degree 2 model y2 + y(fx2 + gxz + hz2) - (ax4 + bx3z + cx2z2 + dxz3 + ez4).
A sequence [a, b, c, d, e, f, g, h, i, j] of length 10 is interpreted as the degree 3 model ax3 + by3 + cz3 + dx2y + ex2z + fy2x + gy2z + hz2x + iz2y + 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.
A genus one model that represents the given curve C.For degree 2, C should either be a subscheme of a weighted projective space 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 P3 for n=4, or an intersection of five quadrics in P4 for n=5.
The genus one model given by the polynomial f or the sequence of equations seq.
A genus one model of degree n (where n is 2, 3, 4, or 5) representing the elliptic curve E embedded in Pn - 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.
The genus one model of degree 5 associated to the given 5 x 5 matrix.
The genus one model of degree 4 determined by the given pair of 4 x 4 symmetric matrices in the sequence mats. (The matrices can be recovered by calling ModelToMatrices).
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.
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.
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.
The genus one model defined over the ring R obtained by coercing the coefficients of the given genus one model into R.
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.
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.
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.
A genus one model of degree n invariant under the diagonal action of μn. The second argument should be a sequence of n ring elements.
> 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]From this matrix, which is the data storing the model, the equations of the curve in P4 can be computed; they are quadratic forms given by the 4 x 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
]
Note that the degree 5 model has the same invariants c4, c6, Δ
as Ea, 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