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This section contains functions for obtaining a simpler model
of a given genus one curve.
We use the following terminology, which has become fairly standard.
Here a model will always mean a global integral model.
Minimisation refers to computing a model which is isomorphic
to the original one (over its ground field) but possibly not integrally
equivalent. A model is minimal if it is locally minimal at all
primes; a model is locally minimal if the valuation of its discriminant
is as small as possible among all integral models. For locally solvable
models this minimal discriminant is equal to that of the associated
elliptic curve.
Reduction refers to computing a model which is integrally
equivalent to the original one
(meaning there exist transformations in both directions over the
ring of integers of the base field). Informally, a
reduced model is one whose coefficients have small height.
One approach to formulating this precisely is to attach to each model
an invariant in some symmetric space: The model is then said
to be reduced iff its invariant lies in some fixed fundamental domain.
This idea is also the basis for algorithms used to obtain reduced models.
The algorithms used for models of degree 2, 3, and 4 over Q
are described in [CFS10]. For degree 5
it is necessary to use some different techniques developed by Fisher.
For models of degree 2 over number fields the algorithms use
additional techniques developed by Donnelly and Fisher.
Transformation: BoolElt Default: true
CrossTerms: BoolElt Default: false
UsePrimes: SeqEnum Default: []
ClassGroupPrimes: SeqEnum Default: []
SetVerbose("Minimisation", n): Maximum: 3
Given a genus one model (of degree 2, 3, 4, or 5) with coefficients in Q,
this returns a minimal model in the sense described above.
It is also implemented for genus one models of degree 2 without cross terms,
and returns a model that is minimal among models without cross terms
when this is possible. When this is not possible (due to class group
obstructions) it returns a model that is nearly minimal, in the sense
that the extra factor appearing in the discriminant is chosen to have
small norm.
The primes dividing the extra factor may be specified via the optional argument
ClassGroupPrimes.
The transformation taking the original model to the minimal model is
also returned, unless the optional argument Transformation is set
to false.
(The syntax for transformations is explained in Section Transformations between Genus One Models).
For degree 2, when the optional argument CrossTerms is set to false
then a model without cross terms is returned that is minimal among models
of this kind.
The third value returned is the set of primes where the minimal model
has positive level.
These are the primes where the model is not soluble over Qpnr,
the maximal non-ramified extension of Qp
(except for p = 2 for models of degree 2 when CrossTerms is false).
The degree 5 routine is not yet proven to work in all cases.
Given the equation f of a nonsingular projective plane cubic curve,
this returns the equation of a minimal model of the curve and a tuple
specifying the transformation
(as explained in Section Transformations between Genus One Models below).
Given the equation f of a nonsingular projective plane cubic curve,
this function returns a model of the curve which is minimal at p.
Also returned is a matrix M giving the transformation; up to scaling,
the minimised cubic is the original cubic evaluated at M[x, y, z]tr.
Reduce(f) : RngMPolElt -> RngMPolElt, TransG1
SetVerbose("Reduce", n): Maximum: 3
Given a genus one model of degree 2, 3, or 4 over Q,
this function computes a reduced model in the sense described above.
The model may instead be described by the appropriate polynomial f,
which must be either a homogeneous polynomial that defines
a genus one model of degree 2 or 3, or a univariate polynomial that
defines a genus one model of degree 2 without cross terms.
The transformation taking the original model to the reduced model
is also returned. (The syntax for transformations between genus one models
is explained in Section Transformations between Genus One Models).
SetVerbose("Reduce", n): Maximum: 3
This function computes a reduced basis for the space spanned by the
given quadrics (which should be given as a sequence of homogeneous quadratic forms
in variables x1, ..., xn).
This means making a linear change of the
homogeneous coordinates, and also finding a new basis for the space
of forms, with the aim of making the resulting coefficients small.
The second and third objects returned are matrices S and T giving the transformation.
The change of homogeneous coordinate variables is given by S, while T specifies
the change of basis of the space of forms.
The returned forms are therefore obtained by substituting
[x1, ..., xn].S for [x1, ..., xn] in the original forms,
and then applying T to this basis of forms (where T acts from the left).
The current implementation of this routine is not optimal; nevertheless
it is useful in some situations.
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