Finite subgroups of modular abelian varieties¶

Sage can compute with fairly general finite subgroups of modular abelian varieties. Elements of finite order are represented by equivalence classes of elements in \(H_1(A,\QQ)\) modulo \(H_1(A,\ZZ)\). A finite subgroup can be defined by giving generators and via various other constructions. Given a finite subgroup, one can compute generators, as well as the structure as an abstract group. Arithmetic on subgroups is also supported, including adding two subgroups together, checking inclusion, etc.

TODO: Intersection, action of Hecke operators.

AUTHORS:

William Stein (2007-03)

EXAMPLES:

Sage

sage: J = J0(33)
sage: C = J.cuspidal_subgroup()
sage: C
Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33) of dimension 3
sage: C.order()
100
sage: C.gens()
[[(1/10, 0, 1/10, 1/10, 1/10, 3/10)], [(0, 1/5, 1/10, 0, 1/10, 9/10)], [(0, 0, 1/2, 0, 1/2, 1/2)]]
sage: C.0 + C.1
[(1/10, 1/5, 1/5, 1/10, 1/5, 6/5)]
sage: 10*(C.0 + C.1)
[(0, 0, 0, 0, 0, 0)]
sage: G = C.subgroup([C.0 + C.1]); G
Finite subgroup with invariants [10] over QQbar of Abelian variety J0(33) of dimension 3
sage: G.gens()
[[(1/10, 1/5, 1/5, 1/10, 1/5, 1/5)]]
sage: G.order()
10
sage: G <= C
True
sage: G >= C
False

Python

>>> from sage.all import *
>>> J = J0(Integer(33))
>>> C = J.cuspidal_subgroup()
>>> C
Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33) of dimension 3
>>> C.order()
100
>>> C.gens()
[[(1/10, 0, 1/10, 1/10, 1/10, 3/10)], [(0, 1/5, 1/10, 0, 1/10, 9/10)], [(0, 0, 1/2, 0, 1/2, 1/2)]]
>>> C.gen(0) + C.gen(1)
[(1/10, 1/5, 1/5, 1/10, 1/5, 6/5)]
>>> Integer(10)*(C.gen(0) + C.gen(1))
[(0, 0, 0, 0, 0, 0)]
>>> G = C.subgroup([C.gen(0) + C.gen(1)]); G
Finite subgroup with invariants [10] over QQbar of Abelian variety J0(33) of dimension 3
>>> G.gens()
[[(1/10, 1/5, 1/5, 1/10, 1/5, 1/5)]]
>>> G.order()
10
>>> G <= C
True
>>> G >= C
False

Sage Live

J = J0(33)
C = J.cuspidal_subgroup()
C
C.order()
C.gens()
C.0 + C.1
10*(C.0 + C.1)
G = C.subgroup([C.0 + C.1]); G
G.gens()
G.order()
G <= C
G >= C

We make a table of the order of the cuspidal subgroup for the first few levels:

Sage

sage: for N in range(11,40):
....:     print("{} {}".format(N, J0(N).cuspidal_subgroup().order()))
5
1
1
6
8
1
4
1
3
6
8
25
11
8
1
21
9
36
7
192
5
8
100
48
48
12
3
135
56

Python

>>> from sage.all import *
>>> for N in range(Integer(11),Integer(40)):
...     print("{} {}".format(N, J0(N).cuspidal_subgroup().order()))
5
1
1
6
8
1
4
1
3
6
8
25
11
8
1
21
9
36
7
192
5
8
100
48
48
12
3
135
56

Sage Live

for N in range(11,40):
    print("{} {}".format(N, J0(N).cuspidal_subgroup().order()))

class sage.modular.abvar.finite_subgroup.FiniteSubgroup(abvar, field_of_definition=Rational Field)[source]¶

Bases: Module

A finite subgroup of a modular abelian variety.

INPUT:

abvar – a modular abelian variety
field_of_definition – a field over which this group is defined

EXAMPLES:

This is an abstract base class, so there are no instances of this class itself:

Sage

sage: A = J0(37)
sage: G = A.torsion_subgroup(3); G
Finite subgroup with invariants [3, 3, 3, 3] over QQ of Abelian variety J0(37) of dimension 2
sage: type(G)
<class 'sage.modular.abvar.finite_subgroup.FiniteSubgroup_lattice_with_category'>
sage: from sage.modular.abvar.finite_subgroup import FiniteSubgroup
sage: isinstance(G, FiniteSubgroup)
True

Python

>>> from sage.all import *
>>> A = J0(Integer(37))
>>> G = A.torsion_subgroup(Integer(3)); G
Finite subgroup with invariants [3, 3, 3, 3] over QQ of Abelian variety J0(37) of dimension 2
>>> type(G)
<class 'sage.modular.abvar.finite_subgroup.FiniteSubgroup_lattice_with_category'>
>>> from sage.modular.abvar.finite_subgroup import FiniteSubgroup
>>> isinstance(G, FiniteSubgroup)
True

Sage Live

A = J0(37)
G = A.torsion_subgroup(3); G
type(G)
from sage.modular.abvar.finite_subgroup import FiniteSubgroup
isinstance(G, FiniteSubgroup)

Element[source]¶: alias of TorsionPoint

abelian_variety()[source]¶

Return the abelian variety that this is a finite subgroup of.

EXAMPLES:

Sage

sage: J = J0(42)
sage: G = J.rational_torsion_subgroup(); G
Torsion subgroup of Abelian variety J0(42) of dimension 5
sage: G.abelian_variety()
Abelian variety J0(42) of dimension 5

Python

>>> from sage.all import *
>>> J = J0(Integer(42))
>>> G = J.rational_torsion_subgroup(); G
Torsion subgroup of Abelian variety J0(42) of dimension 5
>>> G.abelian_variety()
Abelian variety J0(42) of dimension 5

Sage Live

J = J0(42)
G = J.rational_torsion_subgroup(); G
G.abelian_variety()

exponent()[source]¶

Return the exponent of this finite abelian group.

OUTPUT: integer

EXAMPLES:

Sage

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: G.exponent()
4

Python

>>> from sage.all import *
>>> t = J0(Integer(33)).hecke_operator(Integer(7))
>>> G = t.kernel()[Integer(0)]; G
Finite subgroup with invariants [2, 2, 2, 2, 4, 4] over QQ of
 Abelian variety J0(33) of dimension 3
>>> G.exponent()
4

Sage Live

t = J0(33).hecke_operator(7)
G = t.kernel()[0]; G
G.exponent()

field_of_definition()[source]¶

Return the field over which this finite modular abelian variety subgroup is defined. This is a field over which this subgroup is defined.

EXAMPLES:

Sage

sage: J = J0(42)
sage: G = J.rational_torsion_subgroup(); G
Torsion subgroup of Abelian variety J0(42) of dimension 5
sage: G.field_of_definition()
Rational Field

Python

>>> from sage.all import *
>>> J = J0(Integer(42))
>>> G = J.rational_torsion_subgroup(); G
Torsion subgroup of Abelian variety J0(42) of dimension 5
>>> G.field_of_definition()
Rational Field

Sage Live

J = J0(42)
G = J.rational_torsion_subgroup(); G
G.field_of_definition()

gen(n)[source]¶

Return \(n\)-th generator of self.

EXAMPLES:

Sage

sage: J = J0(23)
sage: C = J.torsion_subgroup(3)
sage: C.gens()
[[(1/3, 0, 0, 0)], [(0, 1/3, 0, 0)], [(0, 0, 1/3, 0)], [(0, 0, 0, 1/3)]]
sage: C.gen(0)
[(1/3, 0, 0, 0)]
sage: C.gen(3)
[(0, 0, 0, 1/3)]
sage: C.gen(4)
Traceback (most recent call last):
...
IndexError: list index out of range

Python

>>> from sage.all import *
>>> J = J0(Integer(23))
>>> C = J.torsion_subgroup(Integer(3))
>>> C.gens()
[[(1/3, 0, 0, 0)], [(0, 1/3, 0, 0)], [(0, 0, 1/3, 0)], [(0, 0, 0, 1/3)]]
>>> C.gen(Integer(0))
[(1/3, 0, 0, 0)]
>>> C.gen(Integer(3))
[(0, 0, 0, 1/3)]
>>> C.gen(Integer(4))
Traceback (most recent call last):
...
IndexError: list index out of range

Sage Live

J = J0(23)
C = J.torsion_subgroup(3)
C.gens()
C.gen(0)
C.gen(3)
C.gen(4)

Negative indices wrap around:

Sage

sage: C.gen(-1)
[(0, 0, 0, 1/3)]

Python

>>> from sage.all import *
>>> C.gen(-Integer(1))
[(0, 0, 0, 1/3)]

Sage Live

C.gen(-1)

gens()[source]¶

Return generators for this finite subgroup.

EXAMPLES: We list generators for several cuspidal subgroups:

Sage

sage: J0(11).cuspidal_subgroup().gens()
[[(0, 1/5)]]
sage: J0(37).cuspidal_subgroup().gens()
[[(0, 0, 0, 1/3)]]
sage: J0(43).cuspidal_subgroup().gens()
[[(0, 1/7, 0, 6/7, 0, 5/7)]]
sage: J1(13).cuspidal_subgroup().gens()
[[(1/19, 0, 9/19, 9/19)], [(0, 1/19, 0, 9/19)]]
sage: J0(22).torsion_subgroup(6).gens()
[[(1/6, 0, 0, 0)], [(0, 1/6, 0, 0)], [(0, 0, 1/6, 0)], [(0, 0, 0, 1/6)]]

Python

>>> from sage.all import *
>>> J0(Integer(11)).cuspidal_subgroup().gens()
[[(0, 1/5)]]
>>> J0(Integer(37)).cuspidal_subgroup().gens()
[[(0, 0, 0, 1/3)]]
>>> J0(Integer(43)).cuspidal_subgroup().gens()
[[(0, 1/7, 0, 6/7, 0, 5/7)]]
>>> J1(Integer(13)).cuspidal_subgroup().gens()
[[(1/19, 0, 9/19, 9/19)], [(0, 1/19, 0, 9/19)]]
>>> J0(Integer(22)).torsion_subgroup(Integer(6)).gens()
[[(1/6, 0, 0, 0)], [(0, 1/6, 0, 0)], [(0, 0, 1/6, 0)], [(0, 0, 0, 1/6)]]

Sage Live

J0(11).cuspidal_subgroup().gens()
J0(37).cuspidal_subgroup().gens()
J0(43).cuspidal_subgroup().gens()
J1(13).cuspidal_subgroup().gens()
J0(22).torsion_subgroup(6).gens()

intersection(other)[source]¶

Return the intersection of the finite subgroups self and other.

INPUT:

other – a finite group

OUTPUT: a finite group

EXAMPLES:

Sage

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)

Python

>>> from sage.all import *
>>> E11a0, E11a1, B = J0(Integer(33))
>>> G = E11a0.torsion_subgroup(Integer(6)); H = E11a0.torsion_subgroup(Integer(9))
>>> G.intersection(H)
Finite subgroup with invariants [3, 3] over QQ of
 Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
>>> W = E11a1.torsion_subgroup(Integer(15))
>>> G.intersection(W)
Finite subgroup with invariants [] over QQ of
 Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
>>> E11a0.intersection(E11a1)[Integer(0)]
Finite subgroup with invariants [5] over QQ of
 Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)

Sage Live

E11a0, E11a1, B = J0(33)
G = E11a0.torsion_subgroup(6); H = E11a0.torsion_subgroup(9)
G.intersection(H)
W = E11a1.torsion_subgroup(15)
G.intersection(W)
E11a0.intersection(E11a1)[0]

We intersect subgroups of different abelian varieties.

Sage

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)

Python

>>> from sage.all import *
>>> E11a0, E11a1, B = J0(Integer(33))
>>> G = E11a0.torsion_subgroup(Integer(5)); H = E11a1.torsion_subgroup(Integer(5))
>>> G.intersection(H)
Finite subgroup with invariants [5] over QQ of
 Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
>>> E11a0.intersection(E11a1)[Integer(0)]
Finite subgroup with invariants [5] over QQ of
 Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)

Sage Live

E11a0, E11a1, B = J0(33)
G = E11a0.torsion_subgroup(5); H = E11a1.torsion_subgroup(5)
G.intersection(H)
E11a0.intersection(E11a1)[0]

We intersect abelian varieties with subgroups:

Sage

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)

Python

>>> from sage.all import *
>>> t = J0(Integer(33)).hecke_operator(Integer(7))
>>> G = t.kernel()[Integer(0)]; G
Finite subgroup with invariants [2, 2, 2, 2, 4, 4] over QQ of
 Abelian variety J0(33) of dimension 3
>>> A = J0(Integer(33)).old_subvariety()
>>> A.intersection(G)
Finite subgroup with invariants [2, 2, 2, 2] over QQ of
 Abelian subvariety of dimension 2 of J0(33)
>>> A.hecke_operator(Integer(7)).kernel()[Integer(0)]
Finite subgroup with invariants [2, 2, 2, 2] over QQ of
 Abelian subvariety of dimension 2 of J0(33)
>>> B = J0(Integer(33)).new_subvariety()
>>> B.intersection(G)
Finite subgroup with invariants [4, 4] over QQ of
 Abelian subvariety of dimension 1 of J0(33)
>>> B.hecke_operator(Integer(7)).kernel()[Integer(0)]
Finite subgroup with invariants [4, 4] over QQ of
 Abelian subvariety of dimension 1 of J0(33)
>>> A.intersection(B)[Integer(0)]
Finite subgroup with invariants [3, 3] over QQ of
 Abelian subvariety of dimension 2 of J0(33)

Sage Live

t = J0(33).hecke_operator(7)
G = t.kernel()[0]; G
A = J0(33).old_subvariety()
A.intersection(G)
A.hecke_operator(7).kernel()[0]
B = J0(33).new_subvariety()
B.intersection(G)
B.hecke_operator(7).kernel()[0]
A.intersection(B)[0]

invariants()[source]¶

Return elementary invariants of this abelian group, by which we mean a nondecreasing (immutable) sequence of integers \(n_i\), \(1 \leq i \leq k\), with \(n_i\) dividing \(n_{i+1}\), and such that this group is abstractly isomorphic to \(\ZZ/n_1\ZZ \times\cdots\times \ZZ/n_k\ZZ.\)

EXAMPLES:

Sage

sage: J = J0(38)
sage: C = J.cuspidal_subgroup(); C
Finite subgroup with invariants [3, 45] over QQ of
 Abelian variety J0(38) of dimension 4
sage: v = C.invariants(); v
[3, 45]
sage: v[0] = 5
Traceback (most recent call last):
...
ValueError: object is immutable; please change a copy instead.
sage: type(v[0])
<class 'sage.rings.integer.Integer'>

Python

>>> from sage.all import *
>>> J = J0(Integer(38))
>>> C = J.cuspidal_subgroup(); C
Finite subgroup with invariants [3, 45] over QQ of
 Abelian variety J0(38) of dimension 4
>>> v = C.invariants(); v
[3, 45]
>>> v[Integer(0)] = Integer(5)
Traceback (most recent call last):
...
ValueError: object is immutable; please change a copy instead.
>>> type(v[Integer(0)])
<class 'sage.rings.integer.Integer'>

Sage Live

J = J0(38)
C = J.cuspidal_subgroup(); C
v = C.invariants(); v
v[0] = 5
type(v[0])

Sage

sage: C * 3
Finite subgroup with invariants [15] over QQ of
 Abelian variety J0(38) of dimension 4

Python

>>> from sage.all import *
>>> C * Integer(3)
Finite subgroup with invariants [15] over QQ of
 Abelian variety J0(38) of dimension 4

Sage Live

C * 3

Python

>>> from sage.all import *
>>> C * Integer(3)
Finite subgroup with invariants [15] over QQ of
 Abelian variety J0(38) of dimension 4

Sage Live

C * 3

An example involving another cuspidal subgroup:

Sage

sage: C = J0(22).cuspidal_subgroup(); C
Finite subgroup with invariants [5, 5] over QQ of
 Abelian variety J0(22) of dimension 2
sage: C.lattice()
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[1/5 1/5 4/5   0]
[  0   1   0   0]
[  0   0   1   0]
[  0   0   0 1/5]
sage: C.invariants()
[5, 5]

Python

>>> from sage.all import *
>>> C = J0(Integer(22)).cuspidal_subgroup(); C
Finite subgroup with invariants [5, 5] over QQ of
 Abelian variety J0(22) of dimension 2
>>> C.lattice()
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[1/5 1/5 4/5   0]
[  0   1   0   0]
[  0   0   1   0]
[  0   0   0 1/5]
>>> C.invariants()
[5, 5]

Sage Live

C = J0(22).cuspidal_subgroup(); C
C.lattice()
C.invariants()

is_subgroup(other)[source]¶

Return True exactly if self is a subgroup of other, and both are defined as subgroups of the same ambient abelian variety.

EXAMPLES:

Sage

sage: C = J0(22).cuspidal_subgroup()
sage: H = C.subgroup([C.0])
sage: K = C.subgroup([C.1])
sage: H.is_subgroup(K)
False
sage: K.is_subgroup(H)
False
sage: K.is_subgroup(C)
True
sage: H.is_subgroup(C)
True

Python

>>> from sage.all import *
>>> C = J0(Integer(22)).cuspidal_subgroup()
>>> H = C.subgroup([C.gen(0)])
>>> K = C.subgroup([C.gen(1)])
>>> H.is_subgroup(K)
False
>>> K.is_subgroup(H)
False
>>> K.is_subgroup(C)
True
>>> H.is_subgroup(C)
True

Sage Live

C = J0(22).cuspidal_subgroup()
H = C.subgroup([C.0])
K = C.subgroup([C.1])
H.is_subgroup(K)
K.is_subgroup(H)
K.is_subgroup(C)
H.is_subgroup(C)

lattice()[source]¶

Return the lattice corresponding to this subgroup in the rational homology of the modular Jacobian product. The elements of the subgroup are represented by vectors in the ambient vector space (the rational homology), and this returns the lattice they span. EXAMPLES:

Sage

sage: J = J0(33); C = J[0].cuspidal_subgroup(); C
Finite subgroup with invariants [5] over QQ of
 Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
sage: C.lattice()
Free module of degree 6 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1/5 13/5   -2 -4/5    2 -1/5]
[   0    3   -2   -1    2    0]

Python

>>> from sage.all import *
>>> J = J0(Integer(33)); C = J[Integer(0)].cuspidal_subgroup(); C
Finite subgroup with invariants [5] over QQ of
 Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
>>> C.lattice()
Free module of degree 6 and rank 2 over Integer Ring
Echelon basis matrix:
[ 1/5 13/5   -2 -4/5    2 -1/5]
[   0    3   -2   -1    2    0]

Sage Live

J = J0(33); C = J[0].cuspidal_subgroup(); C
C.lattice()

order()[source]¶

Return the order (number of elements) of this finite subgroup.

EXAMPLES:

Sage

sage: J = J0(42)
sage: C = J.cuspidal_subgroup()
sage: C.order()
2304

Python

>>> from sage.all import *
>>> J = J0(Integer(42))
>>> C = J.cuspidal_subgroup()
>>> C.order()
2304

Sage Live

J = J0(42)
C = J.cuspidal_subgroup()
C.order()

subgroup(gens)[source]¶

Return the subgroup of self spanned by the given generators, which must all be elements of self.

EXAMPLES:

Sage

sage: J = J0(23)
sage: G = J.torsion_subgroup(11); G
Finite subgroup with invariants [11, 11, 11, 11] over QQ of
 Abelian variety J0(23) of dimension 2

Python

>>> from sage.all import *
>>> J = J0(Integer(23))
>>> G = J.torsion_subgroup(Integer(11)); G
Finite subgroup with invariants [11, 11, 11, 11] over QQ of
 Abelian variety J0(23) of dimension 2

Sage Live

J = J0(23)
G = J.torsion_subgroup(11); G

We create the subgroup of the 11-torsion subgroup of \(J_0(23)\) generated by the first \(11\)-torsion point:

Sage

sage: H = G.subgroup([G.0]); H
Finite subgroup with invariants [11] over QQbar of
 Abelian variety J0(23) of dimension 2
sage: H.invariants()
[11]

Python

>>> from sage.all import *
>>> H = G.subgroup([G.gen(0)]); H
Finite subgroup with invariants [11] over QQbar of
 Abelian variety J0(23) of dimension 2
>>> H.invariants()
[11]

Sage Live

H = G.subgroup([G.0]); H
H.invariants()

We can also create a subgroup from a list of objects that can be converted into the ambient rational homology:

Sage

sage: H == G.subgroup([[1/11,0,0,0]])
True

Python

>>> from sage.all import *
>>> H == G.subgroup([[Integer(1)/Integer(11),Integer(0),Integer(0),Integer(0)]])
True

Sage Live

H == G.subgroup([[1/11,0,0,0]])

class sage.modular.abvar.finite_subgroup.FiniteSubgroup_lattice(abvar, lattice, field_of_definition=None, check=True)[source]¶

Bases: FiniteSubgroup

A finite subgroup of a modular abelian variety that is defined by a given lattice.

INPUT:

abvar – a modular abelian variety
lattice – a lattice that contains the lattice of abvar
field_of_definition – the field of definition of this finite group scheme
check – boolean (default: True); whether or not to check that lattice contains the abvar lattice

EXAMPLES:

Sage

sage: J = J0(11)
sage: G = J.finite_subgroup([[1/3,0], [0,1/5]]); G
Finite subgroup with invariants [15] over QQbar of
 Abelian variety J0(11) of dimension 1

Python

>>> from sage.all import *
>>> J = J0(Integer(11))
>>> G = J.finite_subgroup([[Integer(1)/Integer(3),Integer(0)], [Integer(0),Integer(1)/Integer(5)]]); G
Finite subgroup with invariants [15] over QQbar of
 Abelian variety J0(11) of dimension 1

Sage Live

J = J0(11)
G = J.finite_subgroup([[1/3,0], [0,1/5]]); G

lattice()[source]¶

Return lattice that defines this finite subgroup.

EXAMPLES:

Sage

sage: J = J0(11)
sage: G = J.finite_subgroup([[1/3,0], [0,1/5]]); G
Finite subgroup with invariants [15] over QQbar of
 Abelian variety J0(11) of dimension 1
sage: G.lattice()
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1/3   0]
[  0 1/5]

Python

>>> from sage.all import *
>>> J = J0(Integer(11))
>>> G = J.finite_subgroup([[Integer(1)/Integer(3),Integer(0)], [Integer(0),Integer(1)/Integer(5)]]); G
Finite subgroup with invariants [15] over QQbar of
 Abelian variety J0(11) of dimension 1
>>> G.lattice()
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1/3   0]
[  0 1/5]

Sage Live

J = J0(11)
G = J.finite_subgroup([[1/3,0], [0,1/5]]); G
G.lattice()