Wrapper class for abelian groups¶
This class is intended as a template for anything in Sage that needs the
functionality of abelian groups. One can create an AdditiveAbelianGroupWrapper
object from any given set of elements in some given parent, as long as an
_add_
method has been defined.
EXAMPLES:
We create a toy example based on the MordellWeil group of an elliptic curve over \(\QQ\):
sage: E = EllipticCurve('30a2')
sage: pts = [E(4,7,1), E(7/4, 11/8, 1), E(3, 2, 1)]
sage: M = AdditiveAbelianGroupWrapper(pts[0].parent(), pts, [3, 2, 2])
sage: M
Additive abelian group isomorphic to Z/3 + Z/2 + Z/2 embedded in Abelian
group of points on Elliptic Curve defined by y^2 + x*y + y = x^3  19*x + 26
over Rational Field
sage: M.gens()
((4 : 7 : 1), (7/4 : 11/8 : 1), (3 : 2 : 1))
sage: 3*M.0
(0 : 1 : 0)
sage: 3000000000000001 * M.0
(4 : 7 : 1)
sage: M == loads(dumps(M)) # known bug, see https://trac.sagemath.org/sage_trac/ticket/11599#comment:7
True
We check that ridiculous operations are being avoided:
sage: from sage.misc.verbose import set_verbose
sage: set_verbose(2, 'additive_abelian_wrapper.py')
sage: 300001 * M.0
verbose 1 (...: additive_abelian_wrapper.py, _discrete_exp) Calling discrete exp on (1, 0, 0)
(4 : 7 : 1)
sage: set_verbose(0, 'additive_abelian_wrapper.py')
Todo
Implement proper blackbox discrete logarithm (using babystep giantstep). The discrete_exp function can also potentially be speeded up substantially via caching.
Think about subgroups and quotients, which probably won’t work in the current implementation – some fiddly adjustments will be needed in order to be able to pass extra arguments to the subquotient’s init method.

class
sage.groups.additive_abelian.additive_abelian_wrapper.
AdditiveAbelianGroupWrapper
(universe, gens, invariants)¶ Bases:
sage.groups.additive_abelian.additive_abelian_group.AdditiveAbelianGroup_fixed_gens
The parent of
AdditiveAbelianGroupWrapperElement

Element
¶ alias of
AdditiveAbelianGroupWrapperElement

generator_orders
()¶ The orders of the generators with which this group was initialised. (Note that these are not necessarily a minimal set of generators.) Generators of infinite order are returned as 0. Compare
self.invariants()
, which returns the orders of a minimal set of generators.EXAMPLES:
sage: V = Zmod(6)**2 sage: G = AdditiveAbelianGroupWrapper(V, [2*V.0, 3*V.1], [3, 2]) sage: G.generator_orders() (3, 2) sage: G.invariants() (6,)

universe
()¶ The ambient group in which this abelian group lives.
EXAMPLES:
sage: G = AdditiveAbelianGroupWrapper(QQbar, [sqrt(QQbar(2)), sqrt(QQbar(3))], [0, 0]) sage: G.universe() Algebraic Field


class
sage.groups.additive_abelian.additive_abelian_wrapper.
AdditiveAbelianGroupWrapperElement
(parent, vector, element=None, check=False)¶ Bases:
sage.groups.additive_abelian.additive_abelian_group.AdditiveAbelianGroupElement
An element of an
AdditiveAbelianGroupWrapper
.
element
()¶ Return the underlying object that this element wraps.
EXAMPLES:
sage: T = EllipticCurve('65a').torsion_subgroup().gen(0) sage: T; type(T) (0 : 0 : 1) <class 'sage.schemes.elliptic_curves.ell_torsion.EllipticCurveTorsionSubgroup_with_category.element_class'> sage: T.element(); type(T.element()) (0 : 0 : 1) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_number_field'>


class
sage.groups.additive_abelian.additive_abelian_wrapper.
UnwrappingMorphism
(domain)¶ Bases:
sage.categories.morphism.Morphism
The embedding into the ambient group. Used by the coercion framework.