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: # needs sage.schemes
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]); 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 (https://github.com/sagemath/sage/issues/11599#comment:7)
True
Todo
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.
AUTHORS:
David Loeffler (2010)
Lorenz Panny (2017):
AdditiveAbelianGroupWrapper.discrete_log()
Lorenz Panny (2023):
AdditiveAbelianGroupWrapper.from_generators()
- class sage.groups.additive_abelian.additive_abelian_wrapper.AdditiveAbelianGroupWrapper(universe, gens, invariants)#
Bases:
AdditiveAbelianGroup_fixed_gens
This class is used to wrap a subgroup of an existing additive abelian group as a new additive abelian group.
EXAMPLES:
sage: G2 = AdditiveAbelianGroupWrapper(Zmod(42), [2], [21]); G2 Additive abelian group isomorphic to Z/21 embedded in Ring of integers modulo 42 sage: G6 = AdditiveAbelianGroupWrapper(Zmod(42), [6], [7]); G6 Additive abelian group isomorphic to Z/7 embedded in Ring of integers modulo 42 sage: G = AdditiveAbelianGroupWrapper(Zmod(42), [21,14,6], [2,3,7]); G Additive abelian group isomorphic to Z/2 + Z/3 + Z/7 embedded in Ring of integers modulo 42 sage: G.invariants() (42,)
sage: AdditiveAbelianGroupWrapper(QQbar, [sqrt(2), sqrt(3)], [0, 0]) # needs sage.rings.number_field sage.symbolic Additive abelian group isomorphic to Z + Z embedded in Algebraic Field
sage: EllipticCurve(GF(419**2), [1,0]).abelian_group() # indirect doctest # needs sage.rings.finite_rings sage.schemes Additive abelian group isomorphic to Z/420 + Z/420 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 419^2
- Element#
alias of
AdditiveAbelianGroupWrapperElement
- discrete_exp(v)#
Given a list (or other iterable) of length equal to the number of generators of this group, compute the element of the ambient group with those exponents in terms of the generators of self.
EXAMPLES:
sage: G = AdditiveAbelianGroupWrapper(QQbar, # needs sage.rings.number_field ....: [sqrt(QQbar(2)), -1], [0, 0]) sage: v = G.discrete_exp([3, 5]); v # needs sage.rings.number_field -0.7573593128807148? sage: v.parent() is QQbar # needs sage.rings.number_field True
This method is an inverse of
discrete_log()
:sage: orders = [2, 2*3, 2*3*5, 2*3*5*7, 2*3*5*7*11] sage: G = AdditiveAbelianGroup(orders) sage: A = AdditiveAbelianGroupWrapper(G.0.parent(), G.gens(), orders) sage: el = A.random_element() sage: A.discrete_exp(A.discrete_log(el)) == el True
- discrete_log(x, gens=None)#
Given an element of the ambient group, attempt to express it in terms of the generators of this group or the given generators of a subgroup.
ALGORITHM:
This reduces to p-groups, then calls
_discrete_log_pgroup()
which implements a basic version of the recursive algorithm from [Suth2008].AUTHORS:
Lorenz Panny (2017)
EXAMPLES:
sage: G = AdditiveAbelianGroup([2, 2*3, 2*3*5, 2*3*5*7, 2*3*5*7*11]) sage: A = AdditiveAbelianGroupWrapper(G.0.parent(), G.gens(), ....: [g.order() for g in G.gens()]) sage: A.discrete_log(A.discrete_exp([1,5,23,127,539])) (1, 5, 23, 127, 539)
sage: x = polygen(ZZ, 'x') sage: F.<t> = GF(1009**2, modulus=x**2+11); E = EllipticCurve(j=F(940)) # needs sage.rings.finite_rings sage.schemes sage: P, Q = E(900*t + 228, 974*t + 185), E(1007*t + 214, 865*t + 802) # needs sage.rings.finite_rings sage.schemes sage: E.abelian_group().discrete_log(123 * P + 777 * Q, [P, Q]) # needs sage.rings.finite_rings sage.schemes (123, 777)
sage: V = Zmod(8)**2 sage: G = AdditiveAbelianGroupWrapper(V, [[2,2],[4,0]], [4, 2]) sage: G.discrete_log(V([6, 2])) (1, 1) sage: G.discrete_log(V([6, 4])) Traceback (most recent call last): ... ValueError: not in group
sage: G = AdditiveAbelianGroupWrapper(QQbar, [sqrt(2)], [0]) # needs sage.rings.number_field sage.symbolic sage: G.discrete_log(QQbar(2*sqrt(2))) # needs sage.rings.number_field sage.symbolic Traceback (most recent call last): ... NotImplementedError: No black-box discrete log for infinite abelian groups
- static from_generators(gens, universe=None)#
This method constructs the subgroup generated by a sequence of finite-order elements in an additive abelian group.
The elements need not be independent, hence this can be used to perform tasks such as finding relations between some given elements of an abelian group, computing the structure of the generated subgroup, enumerating all elements of the subgroup, and solving discrete-logarithm problems.
EXAMPLES:
sage: G = AdditiveAbelianGroup([15, 30, 45]) sage: gs = [G((1,2,3)), G((4,5,6)), G((7,7,7)), G((3,2,1))] sage: H = AdditiveAbelianGroupWrapper.from_generators(gs); H Additive abelian group isomorphic to Z/90 + Z/15 embedded in Additive abelian group isomorphic to Z/15 + Z/30 + Z/45 sage: H.gens() ((12, 13, 14), (1, 26, 21))
- 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,)
- torsion_subgroup(n=None)#
Return the \(n\)-torsion subgroup of this additive abelian group when \(n\) is given, and the torsion subgroup otherwise.
The [\(n\)-]torsion subgroup consists of all elements whose order is finite [and divides \(n\)].
EXAMPLES:
sage: ords = [2, 2*3, 2*3*5, 0, 2*3*5*7, 2*3*5*7*11] sage: G = AdditiveAbelianGroup(ords) sage: A = AdditiveAbelianGroupWrapper(G.0.parent(), G.gens(), ords) sage: T = A.torsion_subgroup(5) sage: T Additive abelian group isomorphic to Z/5 + Z/5 + Z/5 embedded in Additive abelian group isomorphic to Z/2 + Z/6 + Z/30 + Z + Z/210 + Z/2310 sage: T.gens() ((0, 0, 6, 0, 0, 0), (0, 0, 0, 0, 42, 0), (0, 0, 0, 0, 0, 462))
sage: # needs sage.rings.finite_rings sage.schemes sage: E = EllipticCurve(GF(487^2), [311,205]) sage: T = E.abelian_group().torsion_subgroup(42); T Additive abelian group isomorphic to Z/42 + Z/6 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 311*x + 205 over Finite Field in z2 of size 487^2 sage: [P.order() for P in T.gens()] [42, 6]
sage: # needs sage.schemes sage: E = EllipticCurve('574i1') sage: pts = [E(103,172), E(61,18)] sage: assert pts[0].order() == 7 and pts[1].order() == infinity sage: M = AdditiveAbelianGroupWrapper(pts[0].parent(), pts, [7,0]); M Additive abelian group isomorphic to Z/7 + Z embedded in Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 19353*x + 958713 over Rational Field sage: M.torsion_subgroup() Additive abelian group isomorphic to Z/7 embedded in Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 19353*x + 958713 over Rational Field sage: M.torsion_subgroup(7) Additive abelian group isomorphic to Z/7 embedded in Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 19353*x + 958713 over Rational Field sage: M.torsion_subgroup(5) Trivial group embedded in Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 19353*x + 958713 over Rational Field
AUTHORS:
Lorenz Panny (2022)
- universe()#
The ambient group in which this abelian group lives.
EXAMPLES:
sage: G = AdditiveAbelianGroupWrapper(QQbar, # needs sage.rings.number_field ....: [sqrt(QQbar(2)), sqrt(QQbar(3))], ....: [0, 0]) sage: G.universe() # needs sage.rings.number_field Algebraic Field
- class sage.groups.additive_abelian.additive_abelian_wrapper.AdditiveAbelianGroupWrapperElement(parent, vector, element=None, check=False)#
Bases:
AdditiveAbelianGroupElement
An element of an
AdditiveAbelianGroupWrapper
.- element()#
Return the underlying object that this element wraps.
EXAMPLES:
sage: T = EllipticCurve('65a').torsion_subgroup().gen(0) # needs sage.schemes sage: T; type(T) # needs sage.schemes (0 : 0 : 1) <class 'sage.schemes.elliptic_curves.ell_torsion.EllipticCurveTorsionSubgroup_with_category.element_class'> sage: T.element(); type(T.element()) # needs sage.schemes (0 : 0 : 1) <class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_number_field'>
- class sage.groups.additive_abelian.additive_abelian_wrapper.UnwrappingMorphism(domain)#
Bases:
Morphism
The embedding into the ambient group. Used by the coercion framework.
- sage.groups.additive_abelian.additive_abelian_wrapper.basis_from_generators(gens, ords=None)#
Given a generating set of some finite abelian group (additively written), compute and return a basis of the group.
Note
A basis of a finite abelian group is a generating set \(\{g_1, \ldots, g_n\}\) such that each element of the group can be written as a unique linear combination \(\alpha_1 g_1 + \cdots + \alpha_n g_n\) with each \(\alpha_i \in \{0, \ldots, \mathrm{ord}(g_i)-1\}\).
ALGORITHM: [Suth2007], Algorithm 9.1 & Remark 9.1
EXAMPLES:
sage: # needs sage.groups sage.rings.finite_rings sage: from sage.groups.additive_abelian.additive_abelian_wrapper import basis_from_generators sage: E = EllipticCurve(GF(31337^6,'a'), j=37) sage: E.order() 946988065073788930380545280 sage: (R,S), (ordR,ordS) = basis_from_generators(E.gens()) sage: ordR, ordS (313157428926517503432720, 3024) sage: R.order() == ordR True sage: S.order() == ordS True sage: ordR * ordS == E.order() True sage: R.weil_pairing(S, ordR).multiplicative_order() == ordS True sage: E.abelian_group().invariants() (3024, 313157428926517503432720)