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GenusOneModel(seq) : [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.
GenusOneModel(f) : RngMPolElt -> 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,
and also maps of schemes E to C and C to E.
The genus one model of degree 5 associated to the given 5 by 5 matrix.
The genus one model of degree 4 determined by the given
pair of 4 by 4 symmetric matrices in the sequence mats.
(The matrices can be recovered by calling ModelToMatrices).
Simplifies a genus one model of degree 2 by completing the square
of the multivariate polynomial f.
RandomModel(n) : RngIntElt -> ModelG1
Size: RngIntElt Default:
A random genus one model of degree n, where n is 2, 3, 4 or 5.
The generic genus one model of degree n, where n is 2, 3, 4 or 5.
(The coefficients are indeterminates in a polynomial ring).
The genus one model defined over the ring B obtained by coercing the
coefficients of the given genus one model into B.
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 Ea, b : y2 = x3 + ax + b over Q(a, b).
The model is the image of Ea, b under the embedding in P4
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 P4 can be computed
(they are quadratic forms, given by
the 4 by 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, Δas 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
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