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 Mordell-Weil 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 http://trac.sagemath.org/sage_trac/ticket/11599#comment:7
True


We check that ridiculous operations are being avoided:

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)


TODO:

• Implement proper black-box discrete logarithm (using baby-step giant-step). 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.

The parent of AdditiveAbelianGroupWrapperElement

Element
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.

EXAMPLE:

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.

EXAMPLE:

sage: G = AdditiveAbelianGroupWrapper(QQbar, [sqrt(QQbar(2)), sqrt(QQbar(3))], [0, 0])
sage: G.universe()
Algebraic Field


An element of an AdditiveAbelianGroupWrapper.

element()

Return the underlying object that this element wraps.

EXAMPLE:

sage: T = EllipticCurve('65a').torsion_subgroup().gen(0)
sage: T; type(T)
(0 : 0 : 1)
sage: T.element(); type(T.element())
(0 : 0 : 1)
<class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_number_field'>