def __add__(self, other):
"""
Return the sum of two subgroups.
EXAMPLES::
sage: C = J0(22).cuspidal_subgroup()
sage: C.gens()
[[(1/5, 1/5, 4/5, 0)], [(0, 0, 0, 1/5)]]
sage: A = C.subgroup([C.0]); B = C.subgroup([C.1])
sage: A + B == C
True
"""
if not isinstance(other, FiniteSubgroup):
raise TypeError, "only addition of two finite subgroups is defined"
A = self.abelian_variety()
B = other.abelian_variety()
if not A.in_same_ambient_variety(B):
raise ValueError(
"self and other must be in the same ambient Jacobian")
K = composite_field(self.field_of_definition(),
other.field_of_definition())
lattice = self.lattice() + other.lattice()
if A != B:
lattice += C.lattice()
return FiniteSubgroup_lattice(self.abelian_variety(),
lattice,
field_of_definition=K)
def __add__(self, other):
"""
Return the sum of two subgroups.
EXAMPLES::
sage: C = J0(22).cuspidal_subgroup()
sage: C.gens()
[[(1/5, 1/5, 4/5, 0)], [(0, 0, 0, 1/5)]]
sage: A = C.subgroup([C.0]); B = C.subgroup([C.1])
sage: A + B == C
True
"""
if not isinstance(other, FiniteSubgroup):
raise TypeError, "only addition of two finite subgroups is defined"
A = self.abelian_variety()
B = other.abelian_variety()
if not A.in_same_ambient_variety(B):
raise ValueError("self and other must be in the same ambient Jacobian")
K = composite_field(self.field_of_definition(), other.field_of_definition())
lattice = self.lattice() + other.lattice()
if A != B:
lattice += C.lattice()
return FiniteSubgroup_lattice(self.abelian_variety(), lattice, field_of_definition=K)
def intersection(self, other):
"""
Return the intersection of the finite subgroups self and other.
INPUT:
- ``other`` - a finite group
OUTPUT: a finite group
EXAMPLES::
sage: E11a0, E11a1, B = J0(33)
sage: G = E11a0.torsion_subgroup(6); H = E11a0.torsion_subgroup(9)
sage: G.intersection(H)
Finite subgroup with invariants [3, 3] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: W = E11a1.torsion_subgroup(15)
sage: G.intersection(W)
Finite subgroup with invariants [] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: E11a0.intersection(E11a1)[0]
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect subgroups of different abelian varieties.
::
sage: E11a0, E11a1, B = J0(33)
sage: G = E11a0.torsion_subgroup(5); H = E11a1.torsion_subgroup(5)
sage: G.intersection(H)
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: E11a0.intersection(E11a1)[0]
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect abelian varieties with subgroups::
sage: t = J0(33).hecke_operator(7)
sage: G = t.kernel()[0]; G
Finite subgroup with invariants [2, 2, 2, 2, 4, 4] over QQ of Abelian variety J0(33) of dimension 3
sage: A = J0(33).old_subvariety()
sage: A.intersection(G)
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
sage: A.hecke_operator(7).kernel()[0]
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
sage: B = J0(33).new_subvariety()
sage: B.intersection(G)
Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
sage: B.hecke_operator(7).kernel()[0]
Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
sage: A.intersection(B)[0]
Finite subgroup with invariants [3, 3] over QQ of Abelian subvariety of dimension 2 of J0(33)
"""
A = self.abelian_variety()
if abelian_variety.is_ModularAbelianVariety(other):
amb = other
B = other
M = B.lattice().scale(Integer(1) / self.exponent())
K = composite_field(self.field_of_definition(), other.base_field())
else:
amb = A
if not isinstance(other, FiniteSubgroup):
raise TypeError, "only addition of two finite subgroups is defined"
B = other.abelian_variety()
if A.ambient_variety() != B.ambient_variety():
raise TypeError, "finite subgroups must be in the same ambient product Jacobian"
M = other.lattice()
K = composite_field(self.field_of_definition(),
other.field_of_definition())
L = self.lattice()
if A != B:
# TODO: This might be way slower than what we could do if
# we think more carefully.
C = A + B
L = L + C.lattice()
M = M + C.lattice()
W = L.intersection(M).intersection(amb.vector_space())
return FiniteSubgroup_lattice(amb, W, field_of_definition=K)
def intersection(self, other):
"""
Return the intersection of the finite subgroups self and other.
INPUT:
- ``other`` - a finite group
OUTPUT: a finite group
EXAMPLES::
sage: E11a0, E11a1, B = J0(33)
sage: G = E11a0.torsion_subgroup(6); H = E11a0.torsion_subgroup(9)
sage: G.intersection(H)
Finite subgroup with invariants [3, 3] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: W = E11a1.torsion_subgroup(15)
sage: G.intersection(W)
Finite subgroup with invariants [] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: E11a0.intersection(E11a1)[0]
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect subgroups of different abelian varieties.
::
sage: E11a0, E11a1, B = J0(33)
sage: G = E11a0.torsion_subgroup(5); H = E11a1.torsion_subgroup(5)
sage: G.intersection(H)
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: E11a0.intersection(E11a1)[0]
Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect abelian varieties with subgroups::
sage: t = J0(33).hecke_operator(7)
sage: G = t.kernel()[0]; G
Finite subgroup with invariants [2, 2, 2, 2, 4, 4] over QQ of Abelian variety J0(33) of dimension 3
sage: A = J0(33).old_subvariety()
sage: A.intersection(G)
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
sage: A.hecke_operator(7).kernel()[0]
Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33)
sage: B = J0(33).new_subvariety()
sage: B.intersection(G)
Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
sage: B.hecke_operator(7).kernel()[0]
Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33)
sage: A.intersection(B)[0]
Finite subgroup with invariants [3, 3] over QQ of Abelian subvariety of dimension 2 of J0(33)
"""
A = self.abelian_variety()
if abelian_variety.is_ModularAbelianVariety(other):
amb = other
B = other
M = B.lattice().scale(Integer(1)/self.exponent())
K = composite_field(self.field_of_definition(), other.base_field())
else:
amb = A
if not isinstance(other, FiniteSubgroup):
raise TypeError, "only addition of two finite subgroups is defined"
B = other.abelian_variety()
if A.ambient_variety() != B.ambient_variety():
raise TypeError, "finite subgroups must be in the same ambient product Jacobian"
M = other.lattice()
K = composite_field(self.field_of_definition(), other.field_of_definition())
L = self.lattice()
if A != B:
# TODO: This might be way slower than what we could do if
# we think more carefully.
C = A + B
L = L + C.lattice()
M = M + C.lattice()
W = L.intersection(M).intersection(amb.vector_space())
return FiniteSubgroup_lattice(amb, W, field_of_definition=K)
