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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 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.
GenusOneModel(f) : RngUPolElt -> ModelG1
GenusOneModel(seq) : [ RngMPolElt ] -> ModelG1
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).
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.
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.
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.
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.
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 Q(a, b).
The model is the image of E a, b under the embedding in 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]
From this matrix, which is the data storing the model,
the equations of the curve in P 4 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 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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