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.
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.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')
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.
AdditiveAbelianGroupWrapper(universe, gens, invariants)¶
The parent of
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.
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,)
The ambient group in which this abelian group lives.
sage: G = AdditiveAbelianGroupWrapper(QQbar, [sqrt(QQbar(2)), sqrt(QQbar(3))], [0, 0]) sage: G.universe() Algebraic Field
AdditiveAbelianGroupWrapperElement(parent, vector, element=None, check=False)¶
An element of an
Return the underlying object that this element wraps.
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'>