|
The function IsGL2Equivalent plays a central role in the isomorphism
testing, and is documented here due to its central role in these computations.
This function returns true if and only if f and g are in the same
GL2(k)-orbit, where k is the coefficient field of their parent, modulo
scalars. The polynomials are considered as homogeneous polynomials of degree
n, where n must be at least 4. The second return value is the sequence
of all matrix entries [a, b, c, d] such that g(x) is a constant times
f((ax + b)/(cx + d)) (cx + d)n.
geometric: BoolElt Default: false
commonfield: BoolElt Default: true
covariant: BoolElt Default: true
Returns a boolean indicating whether a matrix T exists such that the change of
variable induced on f1 by T, f1 * T, is a multiple of f2, as well as a full list
of all such matrices.
If geometric is set to true, then the set of isomorphisms over the
algebraic closure of the base field is returned. If commonfield is set to
false, then the isomorphisms that are returned may be defined over
different fields. Of covariant is set to false, then the
calculation of the isomorphisms is performed by a direct methods instead of
applying the usual covariant reduction.
For more details, see [LRS12].
IsReducedIsomorphicHyperellipticCurves(f1, f2) : RngUPolElt , RngUPolElt -> BoolElt, List
geometric: BoolElt Default: false
commonfield: BoolElt Default: true
covariant: BoolElt Default: true
Returns a boolean indicating whether a matrix T exists that induces an
isomorphism f1(x) -> f2(x) (f1 and f2 resp. define X1 and X2), as well as a
full list of all such matrices.
If geometric is set to true, then the set of isomorphisms over the
algebraic closure of the base field is returned. If commonfield is set to
false, then the isomorphisms that are returned may be defined over
different fields. Of covariant is set to false, then the
calculation of the isomorphisms is performed by a direct methods instead of
applying the usual covariant reduction.
ReducedIsomorphismsOfHyperellipticCurves(f1, f2) : RngUPolElt , RngUPolElt -> List
geometric: BoolElt Default: false
commonfield: BoolElt Default: true
covariant: BoolElt Default: true
Returns a full list of matrices T that induce an isomorphism f1(x) -> f2(x)
(f1 and f2 resp. define X1 and X2).
If geometric is set to true, then the set of isomorphisms over the
algebraic closure of the base field is returned. If commonfield is set to
false, then the isomorphisms that are returned may be defined over
different fields. Of covariant is set to false, then the
calculation of the isomorphisms is performed by a direct methods instead of
applying the usual covariant reduction.
For more details, see [LRS12].
ReducedAutomorphismsOfHyperellipticCurve(f) : RngUPolElt -> List
geometric: BoolElt Default: false
commonfield: BoolElt Default: true
covariant: BoolElt Default: true
Return the automorphism group of the defining polynomial of X, as a full list
of matrices T.
If geometric is set to true, then the set of isomorphisms over the
algebraic closure of the base field is returned. If commonfield is set to
false, then the isomorphisms that are returned may be defined over
different fields. Of covariant is set to false, then the
calculation of the isomorphisms is performed by a direct methods instead of
applying the usual covariant reduction.
ReducedAutomorphismGroupOfHyperellipticCurve(f, Autos) : RngUPolElt , List -> GrpPerm, Map
explicit: BoolElt Default: false
Return the automorphisms group defined by the sequence Autos, as a permutation
group (and its representation if explicit is set to true).
ReducedAutomorphismGroupOfHyperellipticCurve(f) : RngUPolElt -> GrpPerm, Map
Return the automorphisms group of the curve y2 = f(x), as a permutation
group (and its representation if explicit is set to true).
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|