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The curve defined by a genus one model of degree n is a principal homogeneous space
for some elliptic curve (namely the Jacobian of the curve).
The data of the
Jacobian, and the covering map of degree n2, can be read from
the invariants and covariants of the model.
Any two models with the same Jacobian can be added together,
as elements of the Weil-Chatelet group.
Below are functions for
adding two models of degree 3, and for "doubling" models
of degree 4 or 5.
A related function for degree 3 models is
ThreeSelmerElement (see Section Three-Descent).
For degree 4 models, the maps can also be computed using
AssociatedEllipticCurve and AssociatedHyperellipticCurve
from the package on four descent (see Section Four-Descent).
Jacobian(C) : Crv -> CrvEll
The Jacobian, returned as an elliptic curve, of the given genus one model,
or of the curve C corresponding to a genus one model.
E: CrvEll Default:
The covering map from the given genus one model to its Jacobian.
Three values are returned: the curve C of degree n corresponding
to the given model, its Jacobian as an elliptic curve E,
and a map of schemes C to E.
If an elliptic curve E is given, it must be isomorphic to the Jacobian,
and then this curve will be the image of the map.
AddCubics(model1, model2 : parameters) : ModelG1, ModelG1 -> ModelG1
model1 + model2 : ModelG1, ModelG1 -> ModelG1
E: CrvEll Default:
ReturnBoth: BoolElt Default: false
Given two ternary cubic polynomials,
or two genus one models of degree 3, that both have the same invariants,
the function computes the sum of the corresponding elements
of H1(Q, E[3]).
If the two cubics do not belong to the same elliptic curve E,
an error results.
See Section Three-Descent for more information about AddCubics.
Given a genus one model of degree 4 or 5, this function
computes twice the associated element in the Weil-Chatelet group,
and returns this as a genus one model (which will have
degree 2 or 5, respectively).
FourToTwoCovering(C : parameters) : Crv -> Crv, Crv, MapSch
C2: Crv Default:
Given a genus one model M of degree 4, or associated curve,
this function returns
three values: the curve C4 in P3 given by M, a plane
quartic curve C2 representing two times M
(in the Weil-Chatelet group), and the map of schemes C4 to C2.
The same information is provided by AssociatedHyperellipticCurve(Curve(model)).
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