# Arithmetic subgroups defined by permutations of cosets¶

A subgroup of finite index $$H$$ of a finitely generated group $$G$$ is completely described by the action of a set of generators of $$G$$ on the right cosets $$H \backslash G = \{Hg\}_{g \in G}$$. After some arbitrary choice of numbering one can identify the action of generators as elements of a symmetric group acting transitively (and satisfying the relations of the relators in G). As $$\SL_2(\ZZ)$$ has a very simple presentation as a central extension of a free product of cyclic groups, one can easily design algorithms from this point of view.

The generators of $$\SL_2(\ZZ)$$ used in this module are named as follows $$s_2$$, $$s_3$$, $$l$$, $$r$$ which are defined by

$\begin{split}s_2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},\quad s_3 = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix},\quad l = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix},\quad r = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}.\end{split}$

Those generators satisfy the following relations

$s_2^2 = s_3^3 = -1, \quad r = s_2^{-1}\ l^{-1}\ s_2.$

In particular not all four are needed to generate the whole group $$\SL_2(\ZZ)$$. Three couples which generate $$\SL_2(\ZZ)$$ are of particular interest:

• $$(l,r)$$ as they are also semigroup generators for the semigroup of matrices in $$\SL_2(\ZZ)$$ with non-negative entries,

• $$(l,s_2)$$ as they are closely related to the continued fraction algorithm,

• $$(s_2,s_3)$$ as the group $$\PSL_2(\ZZ)$$ is the free product of the finite cyclic groups generated by these two elements.

Part of these functions are based on Chris Kurth’s KFarey package [Kur2008]. For tests see the file sage.modular.arithgroup.tests.

REFERENCES:

Todo

• modular Farey symbols

• computation of generators of a modular subgroup with a standard surface group presentation. In other words, compute a presentation of the form

$\langle x_i,y_i,c_j |\ \prod_i [x_i,y_i] \prod_j c_j^{\nu_j} = 1\rangle$

where the elements $$x_i$$ and $$y_i$$ are hyperbolic and $$c_j$$ are parabolic ($$\nu_j=\infty$$) or elliptic elements ($$\nu_j < \infty$$).

• computation of centralizer.

• generation of modular (even) subgroups of fixed index.

AUTHORS:

• Chris Kurth (2008): created KFarey package

• David Loeffler (2009): adapted functions from KFarey for inclusion into Sage

• Vincent Delecroix (2010): implementation for odd groups, new design, improvements, documentation

• David Loeffler (2011): congruence testing for odd subgroups, enumeration of liftings of projective subgroups

• David Loeffler & Thomas Hamilton (2012): generalised Hsu congruence test for odd subgroups

sage.modular.arithgroup.arithgroup_perm.ArithmeticSubgroup_Permutation(L=None, R=None, S2=None, S3=None, relabel=False, check=True)

Construct a subgroup of $$\SL_2(\ZZ)$$ from the action of generators on its right cosets.

Return an arithmetic subgroup knowing the action, given by permutations, of at least two standard generators on the its cosets. The generators considered are the following matrices:

$\begin{split}s_2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},\quad s_3 = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix},\quad l = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix},\quad r = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}.\end{split}$

An error will be raised if only one permutation is given. If no arguments are given at all, the full modular group $$\SL(2, \ZZ)$$ is returned.

INPUT:

• S2, S3, L, R - permutations - action of matrices on the right cosets (each coset is identified to an element of $$\{1,\dots,n\}$$ where $$1$$ is reserved for the identity coset).

• relabel - boolean (default: False) - if True, renumber the cosets in a canonical way.

• check - boolean (default: True) - check that the input is valid (it may be time efficient but less safe to set it to False)

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)(3,4)",S3="(1,2,3)")
sage: G
Arithmetic subgroup with permutations of right cosets
S2=(1,2)(3,4)
S3=(1,2,3)
L=(1,4,3)
R=(2,4,3)
sage: G.index()
4

sage: G = ArithmeticSubgroup_Permutation(); G
Arithmetic subgroup with permutations of right cosets
S2=()
S3=()
L=()
R=()
sage: G == SL2Z
True


Some invalid inputs:

sage: ArithmeticSubgroup_Permutation(S2="(1,2)")
Traceback (most recent call last):
...
ValueError: Need at least two generators
sage: ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(3,4,5)")
Traceback (most recent call last):
...
ValueError: Permutations do not generate a transitive group
sage: ArithmeticSubgroup_Permutation(L="(1,2)",R="(1,2,3)")
Traceback (most recent call last):
...
ValueError: Wrong relations between generators
sage: ArithmeticSubgroup_Permutation(S2="(1,2,3,4)",S3="()")
Traceback (most recent call last):
...
ValueError: S2^2 does not equal to S3^3
sage: ArithmeticSubgroup_Permutation(S2="(1,4,2,5,3)", S3="(1,3,5,2,4)")
Traceback (most recent call last):
...
ValueError: S2^2 = S3^3 must have order 1 or 2


The input checks can be disabled for speed:

sage: ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(3,4,5)", check=False) # don't do this!
Arithmetic subgroup with permutations of right cosets
S2=(1,2)
S3=(3,4,5)
L=(1,2)(3,5,4)
R=(1,2)(3,4,5)

class sage.modular.arithgroup.arithgroup_perm.ArithmeticSubgroup_Permutation_class

A subgroup of $$\SL_2(\ZZ)$$ defined by the action of generators on its coset graph.

The class stores the action of generators $$s_2, s_3, l, r$$ on right cosets $$Hg$$ of a finite index subgroup $$H < \SL_2(\ZZ)$$. In particular the action of $$\SL_2(\ZZ)$$ on the cosets is on right.

$\begin{split}s_2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},\quad s_3 = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix},\quad l = \begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix},\quad r = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}.\end{split}$
L()

Return the action of the matrix $$l$$ as a permutation of cosets.

$\begin{split}l = \begin{pmatrix}1&1\\0&1\end{pmatrix}\end{split}$

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(1,2,3)")
sage: G.L()
(1,3)

R()

Return the action of the matrix $$r$$ as a permutation of cosets.

$\begin{split}r = \begin{pmatrix}1&0\\1&1\end{pmatrix}\end{split}$

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(1,2,3)")
sage: G.R()
(2,3)

S2()

Return the action of the matrix $$s_2$$ as a permutation of cosets.

$\begin{split}s_2 = \begin{pmatrix}0&-1\\1&0\end{pmatrix}\end{split}$

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(1,2,3)")
sage: G.S2()
(1,2)

S3()

Return the action of the matrix $$s_3$$ as a permutation of cosets.

$\begin{split}s_3 = \begin{pmatrix} 0 & 1 \\ -1 & 1 \end{pmatrix},\quad\end{split}$

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)",S3="(1,2,3)")
sage: G.S3()
(1,2,3)

congruence_closure()

Returns the smallest congruence subgroup containing self. If self is congruence, this is just self, but represented as a congruence subgroup data type. If self is not congruence, then it may be larger.

In practice, we use the following criterion: let $$m$$ be the generalised level of self. If this subgroup is even, let $$n = m$$, else let $$n = 2m$$. Then any congruence subgroup containing self contains $$\Gamma(n)$$ (a generalisation of Wohlfahrt’s theorem due to Kiming, Verrill and Schuett). So we compute the image of self modulo $$n$$ and return the preimage of that.

Note

If you just want to know if the subgroup is congruence or not, it is much faster to use is_congruence().

EXAMPLES:

sage: Gamma1(3).as_permutation_group().congruence_closure()
Congruence subgroup of SL(2,Z) of level 3, preimage of:
Matrix group over Ring of integers modulo 3 with 2 generators (
[1 1]  [1 2]
[0 1], [0 1]
)
sage: sage.modular.arithgroup.arithgroup_perm.HsuExample10().congruence_closure()  # long time (11s on sage.math, 2012)
Modular Group SL(2,Z)

coset_graph(right_cosets=False, s2_edges=True, s3_edges=True, l_edges=False, r_edges=False, s2_label='s2', s3_label='s3', l_label='l', r_label='r')

Return the right (or left) coset graph.

INPUT:

• right_cosets - bool (default: False) - right or left coset graph

• s2_edges - bool (default: True) - put edges associated to s2

• s3_edges - bool (default: True) - put edges associated to s3

• l_edges - bool (default: False) - put edges associated to l

• r_edges - bool (default: False) - put edges associated to r

• s2_label, s3_label, l_label, r_label - the labels to put on the edges corresponding to the generators action. Use None for no label.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)",S3="()")
sage: G
Arithmetic subgroup with permutations of right cosets
S2=(1,2)
S3=()
L=(1,2)
R=(1,2)
sage: G.index()
2
sage: G.coset_graph()
Looped multi-digraph on 2 vertices

generalised_level()

Return the generalised level of this subgroup.

The generalised level of a subgroup of the modular group is the least common multiple of the widths of the cusps. It was proven by Wohlfart that for even congruence subgroups, the (conventional) level coincides with the generalised level. For odd congruence subgroups the level is either the generalised level, or twice the generalised level [KSV2011].

EXAMPLES:

sage: G = Gamma(2).as_permutation_group()
sage: G.generalised_level()
2
sage: G = Gamma0(3).as_permutation_group()
sage: G.generalised_level()
3

index()

Returns the index of this modular subgroup in the full modular group.

EXAMPLES:

sage: G = Gamma(2)
sage: P = G.as_permutation_group()
sage: P.index()
6
sage: G.index() == P.index()
True

sage: G = Gamma0(8)
sage: P = G.as_permutation_group()
sage: P.index()
12
sage: G.index() == P.index()
True

sage: G = Gamma1(6)
sage: P = G.as_permutation_group()
sage: P.index()
24
sage: G.index() == P.index()
True

is_congruence()

Return True if this is a congruence subgroup, and False otherwise.

ALGORITHM:

Uses Hsu’s algorithm [Hsu1996]. Adapted from Chris Kurth’s implementation in KFarey [Kur2008].

For odd subgroups, Hsu’s algorithm still works with minor modifications, using the extension of Wohlfarht’s theorem due to Kiming, Schuett and Verrill [KSV2011]. See [HL2014] for details.

The algorithm is as follows. Let $$G$$ be a finite-index subgroup of $$\SL(2, \ZZ)$$, and let $$L$$ and $$R$$ be the permutations of the cosets of $$G$$ given by the elements $$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$ and $$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$. Let $$N$$ be the generalized level of $$G$$ (if $$G$$ is even) or twice the generalized level (if $$G$$ is odd). Then:

• if $$N$$ is odd, $$G$$ is congruence if and only if the relation

$(L R^{-1} L)^2 = (R^2 L^{1/2})^3$

holds, where $$1/2$$ is understood as the multiplicative inverse of 2 modulo N.

• if $$N$$ is a power of 2, then $$G$$ is congruence if and only if the relations

$\begin{split}\begin{array}{cc} (L R^{-1} L)^{-1} S (L R^{-1} L) S = 1 & (A1)\\ S^{-1} R S = R^{25} & (A2)\\ (L R^{-1} L)^2 = (S R^5 L R^{-1} L)^3 & (A3) \\ \end{array}\end{split}$

hold, where $$S = L^{20} R^{1/5} L^{-4} R^{-1}$$, $$1/5$$ being the inverse of 5 modulo N.

• if $$N$$ is neither odd nor a power of 2, seven relations (B1-7) hold, for which see [HL2014], or the source code of this function.

If the Sage verbosity flag is set (using set_verbose()), then extra output will be produced indicating which of the relations (A1-3) or (B1-7) is not satisfied.

EXAMPLES:

Test if $$\SL_2(\ZZ)$$ is congruence:

sage: a = ArithmeticSubgroup_Permutation(L='',R='')
sage: a.index()
1
sage: a.is_congruence()
True


This example is congruence – it is $$\Gamma_0(3)$$ in disguise:

sage: S2 = SymmetricGroup(4)
sage: l = S2((2,3,4))
sage: r = S2((1,3,4))
sage: G = ArithmeticSubgroup_Permutation(L=l,R=r)
sage: G
Arithmetic subgroup with permutations of right cosets
S2=(1,2)(3,4)
S3=(1,4,2)
L=(2,3,4)
R=(1,3,4)
sage: G.is_congruence()
True


This one is noncongruence:

sage: import sage.modular.arithgroup.arithgroup_perm as ap
sage: ap.HsuExample10().is_congruence()
False


The following example (taken from [KSV2011]) shows that a lifting of a congruence subgroup of $$\PSL(2,\ZZ)$$ to a subgroup of $$\SL(2, \ZZ)$$ need not necessarily be congruence:

sage: S2 = "(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23)"
sage: S3 = "(1,14,15,13,2,3)(4,5,6,16,17,18)(7,8,9,19,20,21)(10,11,12,22,23,24)"
sage: G = ArithmeticSubgroup_Permutation(S2=S2,S3=S3)
sage: G.is_congruence()
False
sage: G.to_even_subgroup().is_congruence()
True


In fact $$G$$ is a lifting to $$\SL(2,\ZZ)$$ of the group $$\bar{\Gamma}_0(6)$$:

sage: G.to_even_subgroup() == Gamma0(6)
True

is_normal()

Test whether the group is normal

EXAMPLES:

sage: G = Gamma(2).as_permutation_group()
sage: G.is_normal()
True

sage: G = Gamma1(2).as_permutation_group()
sage: G.is_normal()
False

perm_group()

Return the underlying permutation group.

The permutation group returned is isomorphic to the action of the generators $$s_2$$ (element of order two), $$s_3$$ (element of order 3), $$l$$ (parabolic element) and $$r$$ (parabolic element) on right cosets (the action is on the right).

EXAMPLES:

sage: import sage.modular.arithgroup.arithgroup_perm as ap
sage: ap.HsuExample10().perm_group()
Permutation Group with generators [(1,2)(3,4)(5,6)(7,8)(9,10), (1,8,3)(2,4,6)(5,7,10), (1,4)(2,5,9,10,8)(3,7,6), (1,7,9,10,6)(2,3)(4,5,8)]

permutation_action(x)

Given an element x of $$\SL_2(\ZZ)$$, compute the permutation of the cosets of self given by right multiplication by x.

EXAMPLES:

sage: import sage.modular.arithgroup.arithgroup_perm as ap
sage: ap.HsuExample10().permutation_action(SL2Z([32, -21, -67, 44]))
(1,4,6,2,10,5,3,7,8,9)

random_element(initial_steps=30)

Returns a random element in this subgroup.

The algorithm uses a random walk on the Cayley graph of $$\SL_2(\ZZ)$$ stopped at the first time it reaches the subgroup after at least initial_steps steps.

INPUT:

• initial_steps - positive integer (default: 30)

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2='(1,3)(4,5)',S3='(1,2,5)(3,4,6)')
sage: all(G.random_element() in G for _ in range(10))
True

relabel(inplace=True)

Relabel the cosets of this modular subgroup in a canonical way.

The implementation of modular subgroup by action of generators on cosets depends on the choice of a numbering. This function provides canonical labels in the sense that two equal subgroups with different labels are relabeled the same way. The default implementation relabels the group itself. If you want to get a relabeled copy of your modular subgroup, put to False the option inplace.

ALGORITHM:

We give an overview of how the canonical labels for the modular subgroup are built. The procedure only uses the permutations $$S3$$ and $$S2$$ that define the modular subgroup and can be used to renumber any transitive action of the symmetric group. In other words, the algorithm construct a canonical representative of a transitive subgroup in its conjugacy class in the full symmetric group.

1. The identity is still numbered $$0$$ and set the current vertex to be the identity.

2. Number the cycle of $$S3$$ the current vertex belongs to: if the current vertex is labeled $$n$$ then the numbering in such way that the cycle becomes $$(n, n+1, \ldots, n+k)$$).

3. Find a new current vertex using the permutation $$S2$$. If all vertices are relabeled then it’s done, otherwise go to step 2.

EXAMPLES:

sage: S2 = "(1,2)(3,4)(5,6)"; S3 = "(1,2,3)(4,5,6)"
sage: G1 = ArithmeticSubgroup_Permutation(S2=S2,S3=S3); G1
Arithmetic subgroup with permutations of right cosets
S2=(1,2)(3,4)(5,6)
S3=(1,2,3)(4,5,6)
L=(1,4,5,3)
R=(2,4,6,3)
sage: G1.relabel(); G1
Arithmetic subgroup with permutations of right cosets
S2=(1,2)(3,4)(5,6)
S3=(1,2,3)(4,5,6)
L=(1,4,5,3)
R=(2,4,6,3)

sage: S2 = "(1,2)(3,5)(4,6)"; S3 = "(1,2,3)(4,5,6)"
sage: G2 = ArithmeticSubgroup_Permutation(S2=S2,S3=S3); G2
Arithmetic subgroup with permutations of right cosets
S2=(1,2)(3,5)(4,6)
S3=(1,2,3)(4,5,6)
L=(1,5,6,3)
R=(2,5,4,3)
sage: G2.relabel(); G2
Arithmetic subgroup with permutations of right cosets
S2=(1,2)(3,4)(5,6)
S3=(1,2,3)(4,5,6)
L=(1,4,5,3)
R=(2,4,6,3)

sage: S2 = "(1,2)(3,6)(4,5)"; S3 = "(1,2,3)(4,5,6)"
sage: G3 = ArithmeticSubgroup_Permutation(S2=S2,S3=S3); G3
Arithmetic subgroup with permutations of right cosets
S2=(1,2)(3,6)(4,5)
S3=(1,2,3)(4,5,6)
L=(1,6,4,3)
R=(2,6,5,3)
sage: G4 = G3.relabel(inplace=False)
sage: G4
Arithmetic subgroup with permutations of right cosets
S2=(1,2)(3,4)(5,6)
S3=(1,2,3)(4,5,6)
L=(1,4,5,3)
R=(2,4,6,3)
sage: G3 is G4
False

surgroups()

Return an iterator through the non-trivial intermediate groups between $$SL(2,\ZZ)$$ and this finite index group.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)(3,4)(5,6)", S3="(1,2,3)(4,5,6)")
sage: H = next(G.surgroups())
sage: H
Arithmetic subgroup with permutations of right cosets
S2=(1,2)
S3=(1,2,3)
L=(1,3)
R=(2,3)
sage: G.is_subgroup(H)
True


The principal congruence group $$\Gamma(3)$$ has thirteen surgroups:

sage: G = Gamma(3).as_permutation_group()
sage: G.index()
24
sage: l = []
sage: for H in G.surgroups():
....:     l.append(H.index())
....:     assert G.is_subgroup(H) and H.is_congruence()
sage: l
[6, 3, 4, 8, 4, 8, 4, 12, 4, 6, 6, 8, 8]

class sage.modular.arithgroup.arithgroup_perm.EvenArithmeticSubgroup_Permutation(S2, S3, L, R, canonical_labels=False)

An arithmetic subgroup of $$\SL(2, \ZZ)$$ containing $$-1$$, represented in terms of the right action of $$\SL(2, \ZZ)$$ on its cosets.

EXAMPLES:

Construct a noncongruence subgroup of index 7 (the smallest possible):

sage: a2 = SymmetricGroup(7)([(1,2),(3,4),(6,7)]); a3 = SymmetricGroup(7)([(1,2,3),(4,5,6)])
sage: G = ArithmeticSubgroup_Permutation(S2=a2, S3=a3); G
Arithmetic subgroup with permutations of right cosets
S2=(1,2)(3,4)(6,7)
S3=(1,2,3)(4,5,6)
L=(1,4,7,6,5,3)
R=(2,4,5,7,6,3)
sage: G.index()
7
sage: G.dimension_cusp_forms(4)
1
sage: G.is_congruence()
False


Convert some standard congruence subgroups into permutation form:

sage: G = Gamma0(8).as_permutation_group()
sage: G.index()
12
sage: G.is_congruence()
True

sage: G = Gamma0(12).as_permutation_group()
sage: G
Arithmetic subgroup of index 24
sage: G.is_congruence()
True


The following is the unique index 2 even subgroup of $$\SL_2(\ZZ)$$:

sage: w = SymmetricGroup(2)([2,1])
sage: G = ArithmeticSubgroup_Permutation(L=w, R=w)
sage: G.dimension_cusp_forms(6)
1
sage: G.genus()
0

coset_reps()

Return coset representatives.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)(3,4)",S3="(1,2,3)")
sage: c = G.coset_reps()
sage: len(c)
4
sage: [g in G for g in c]
[True, False, False, False]

cusp_widths(exp=False)

Return the list of cusp widths of the group.

EXAMPLES:

sage: G = Gamma(2).as_permutation_group()
sage: G.cusp_widths()
[2, 2, 2]
sage: G.cusp_widths(exp=True)
{2: 3}

sage: S2 = "(1,2)(3,4)(5,6)"
sage: S3 = "(1,2,3)(4,5,6)"
sage: G = ArithmeticSubgroup_Permutation(S2=S2,S3=S3)
sage: G.cusp_widths()
[1, 1, 4]
sage: G.cusp_widths(exp=True)
{1: 2, 4: 1}

sage: S2 = "(1,2)(3,4)(5,6)"
sage: S3 = "(1,3,5)(2,4,6)"
sage: G = ArithmeticSubgroup_Permutation(S2=S2,S3=S3)
sage: G.cusp_widths()

sage: G.cusp_widths(exp=True)
{6: 1}

is_even()

Returns True if this subgroup contains the matrix $$-Id$$.

EXAMPLES:

sage: G = Gamma(2).as_permutation_group()
sage: G.is_even()
True

is_odd()

Returns True if this subgroup does not contain the matrix $$-Id$$.

EXAMPLES:

sage: G = Gamma(2).as_permutation_group()
sage: G.is_odd()
False

ncusps()

Return the number of cusps of this arithmetic subgroup.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2)(3,4)(5,6)",S3="(1,2,3)(4,5,6)")
sage: G.ncusps()
3

nu2()

Returns the number of orbits of elliptic points of order 2 for this arithmetic subgroup.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,4)(2)(3)",S3="(1,2,3)(4)")
sage: G.nu2()
2

nu3()

Returns the number of orbits of elliptic points of order 3 for this arithmetic subgroup.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,4)(2)(3)",S3="(1,2,3)(4)")
sage: G.nu3()
1

odd_subgroups()

Return a list of the odd subgroups of index 2 in $$\Gamma$$, where $$\Gamma$$ is this subgroup. (Equivalently, return the liftings of $$\bar{\Gamma} \le \PSL(2, \ZZ)$$ to $$\SL(2, \ZZ)$$.) This can take rather a long time if the index of this subgroup is large.

one_odd_subgroup(), which returns just one of the odd subgroups (which is much quicker than enumerating them all).

ALGORITHM:

• If $$\Gamma$$ has an element of order 4, then there are no index 2 odd subgroups, so return the empty set.

• If $$\Gamma$$ has no elements of order 4, then the permutation $$S_2$$ is a combination of 2-cycles with no fixed points on $$\{1, \dots, N\}$$. We construct the permutation $$\tilde{S}_2$$ of $$\{1, \dots, 2N\}$$ which has a 4-cycle $$(a, b, a+N, b+N)$$ for each 2-cycle $$(a,b)$$ in S2. Similarly, we construct a permutation $$\tilde{S}_3$$ which has a 6-cycle $$(a,b,c,a+N,b+N,c+N)$$ for each 3-cycle $$(a,b,c)$$ in $$S_3$$, and a 2-cycle $$(a,a+N)$$ for each fixed point $$a$$ of $$S_3$$.

Then the permutations $$\tilde{S}_2$$ and $$\tilde{S}_3$$ satisfy $$\tilde{S}_2^2 = \tilde{S}_3^3 = \iota$$ where $$\iota$$ is the order 2 permutation interchanging $$a$$ and $$a+N$$ for each $$a$$. So the subgroup corresponding to these permutations is an index 2 odd subgroup of $$\Gamma$$.

• The other index 2 odd subgroups of $$\Gamma$$ are obtained from the pairs $$\tilde{S}_2, \tilde{S}_3^\sigma$$ where $$\sigma$$ is an element of the group generated by the 2-cycles $$(a, a+N)$$.

Studying the permutations in the first example below gives a good illustration of the algorithm.

EXAMPLES:

sage: G = sage.modular.arithgroup.arithgroup_perm.HsuExample10()
sage: [G.S2(), G.S3()]
[(1,2)(3,4)(5,6)(7,8)(9,10), (1,8,3)(2,4,6)(5,7,10)]
sage: X = G.odd_subgroups()
sage: for u in X: print([u.S2(), u.S3()])
[(1,2,11,12)(3,4,13,14)(5,6,15,16)(7,8,17,18)(9,10,19,20), (1,8,3,11,18,13)(2,4,6,12,14,16)(5,7,10,15,17,20)(9,19)]
[(1,2,11,12)(3,4,13,14)(5,6,15,16)(7,8,17,18)(9,10,19,20), (1,18,13,11,8,3)(2,4,6,12,14,16)(5,7,10,15,17,20)(9,19)]
[(1,2,11,12)(3,4,13,14)(5,6,15,16)(7,8,17,18)(9,10,19,20), (1,8,13,11,18,3)(2,4,6,12,14,16)(5,7,10,15,17,20)(9,19)]
[(1,2,11,12)(3,4,13,14)(5,6,15,16)(7,8,17,18)(9,10,19,20), (1,18,3,11,8,13)(2,4,6,12,14,16)(5,7,10,15,17,20)(9,19)]


A projective congruence subgroup may have noncongruence liftings, as the example of $$\bar{\Gamma}_0(6)$$ illustrates (see [KSV2011]):

sage: X = Gamma0(6).as_permutation_group().odd_subgroups(); Sequence([[u.S2(), u.S3()] for u in X],cr=True)
[
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,2,3,13,14,15)(4,5,6,16,17,18)(7,8,9,19,20,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,14,15,13,2,3)(4,5,6,16,17,18)(7,8,9,19,20,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,2,3,13,14,15)(4,17,6,16,5,18)(7,8,9,19,20,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,14,15,13,2,3)(4,17,6,16,5,18)(7,8,9,19,20,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,2,3,13,14,15)(4,5,6,16,17,18)(7,20,9,19,8,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,14,15,13,2,3)(4,5,6,16,17,18)(7,20,9,19,8,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,2,3,13,14,15)(4,17,6,16,5,18)(7,20,9,19,8,21)(10,11,12,22,23,24)],
[(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), (1,14,15,13,2,3)(4,17,6,16,5,18)(7,20,9,19,8,21)(10,11,12,22,23,24)]
]
sage: [u.is_congruence() for u in X]
[True, False, False, True, True, False, False, True]

one_odd_subgroup(random=False)

Return an odd subgroup of index 2 in $$\Gamma$$, where $$\Gamma$$ is this subgroup. If the optional argument random is False (the default), this returns an arbitrary but consistent choice from the set of index 2 odd subgroups. If random is True, then it will choose one of these at random.

For details of the algorithm used, see the docstring for the related function odd_subgroups(), which returns a list of all the index 2 odd subgroups.

EXAMPLES:

Starting from $$\Gamma(4)$$ we get back $$\Gamma(4)$$:

sage: G = Gamma(4).as_permutation_group()
sage: G.is_odd(), G.index()
(True, 48)
sage: Ge = G.to_even_subgroup()
sage: Go = Ge.one_odd_subgroup()
sage: Go.is_odd(), Go.index()
(True, 48)
sage: Go == G
True


Strating from $$\Gamma(6)$$ we get a different group:

sage: G = Gamma(6).as_permutation_group()
sage: G.is_odd(), G.index()
(True, 144)
sage: Ge = G.to_even_subgroup()
sage: Go = Ge.one_odd_subgroup()
sage: Go.is_odd(), Go.index()
(True, 144)
sage: Go == G
False


An error will be raised if there are no such subgroups, which occurs if and only if the group contains an element of order 4:

sage: Gamma0(10).as_permutation_group().one_odd_subgroup()
Traceback (most recent call last):
...
ValueError: Group contains an element of order 4, hence no index 2 odd subgroups


Testing randomness:

sage: G = Gamma(6).as_permutation_group().to_even_subgroup()
sage: G1 = G.one_odd_subgroup(random=True) # random
sage: G1.is_subgroup(G)
True

to_even_subgroup(relabel=True)

Return the subgroup generated by self and -Id. Since self is even, this is just self. Provided for compatibility.

EXAMPLES:

sage: G = Gamma0(4).as_permutation_group()
sage: H = G.to_even_subgroup()
sage: H == G
True

todd_coxeter()

Returns a 4-tuple (coset_reps, gens, l, s2) where coset_reps is a list of coset representatives of the subgroup, gens a list of generators, l and s2 are list that corresponds to the action of the matrix $$S2$$ and $$L$$ on the cosets.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2='(1,2)(3,4)',S3='(1,2,3)')
sage: reps,gens,l,s=G.todd_coxeter_l_s2()
sage: reps
[
[1 0]  [ 0 -1]  [1 2]  [1 1]
[0 1], [ 1  0], [0 1], [0 1]
]
sage: len(gens)
3
sage: Matrix(2, 2, [1, 3, 0, 1]) in gens
True
sage: Matrix(2, 2, [1, 0, -1, 1]) in gens
True
sage: Matrix(2, 2, [1, -3, 1, -2]) in gens or Matrix(2, 2, [2, -3, 1, -1]) in gens
True
sage: l
[3, 1, 0, 2]
sage: s
[1, 0, 3, 2]
sage: S2 = SL2Z([0,-1,1,0])
sage: L = SL2Z([1,1,0,1])
sage: reps == SL2Z([1,0,0,1])
True
sage: all(reps[i]*S2*~reps[s[i]] in G for i in range(4))
True
sage: all(reps[i]*L*~reps[l[i]] in G for i in range(4))
True

todd_coxeter_l_s2()

Returns a 4-tuple (coset_reps, gens, l, s2) where coset_reps is a list of coset representatives of the subgroup, gens a list of generators, l and s2 are list that corresponds to the action of the matrix $$S2$$ and $$L$$ on the cosets.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2='(1,2)(3,4)',S3='(1,2,3)')
sage: reps,gens,l,s=G.todd_coxeter_l_s2()
sage: reps
[
[1 0]  [ 0 -1]  [1 2]  [1 1]
[0 1], [ 1  0], [0 1], [0 1]
]
sage: len(gens)
3
sage: Matrix(2, 2, [1, 3, 0, 1]) in gens
True
sage: Matrix(2, 2, [1, 0, -1, 1]) in gens
True
sage: Matrix(2, 2, [1, -3, 1, -2]) in gens or Matrix(2, 2, [2, -3, 1, -1]) in gens
True
sage: l
[3, 1, 0, 2]
sage: s
[1, 0, 3, 2]
sage: S2 = SL2Z([0,-1,1,0])
sage: L = SL2Z([1,1,0,1])
sage: reps == SL2Z([1,0,0,1])
True
sage: all(reps[i]*S2*~reps[s[i]] in G for i in range(4))
True
sage: all(reps[i]*L*~reps[l[i]] in G for i in range(4))
True

todd_coxeter_s2_s3()

Returns a 4-tuple (coset_reps, gens, s2, s3) where coset_reps are coset representatives of the subgroup, gens is a list of generators, s2 and s3 are the action of the matrices $$S2$$ and $$S3$$ on the list of cosets.

The so called Todd-Coxeter algorithm is a general method for coset enumeration for a subgroup of a group given by generators and relations.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2='(1,2)(3,4)',S3='(1,2,3)')
sage: G.genus()
0
sage: reps,gens,s2,s3=G.todd_coxeter_s2_s3()
sage: g1,g2 = gens
sage: g1 in G and g2 in G
True
sage: Matrix(2, 2, [-1, 3, -1, 2]) in gens
True
sage: Matrix(2, 2, [-1, 0, 1, -1]) in gens or Matrix(2, 2, [1, 0, 1, 1]) in gens
True
sage: S2 = SL2Z([0,-1,1,0])
sage: S3 = SL2Z([0,1,-1,1])
sage: reps == SL2Z([1,0,0,1])
True
sage: all(reps[i]*S2*~reps[s2[i]] in G for i in range(4))
True
sage: all(reps[i]*S3*~reps[s3[i]] in G for i in range(4))
True

sage.modular.arithgroup.arithgroup_perm.HsuExample10()

An example of an index 10 arithmetic subgroup studied by Tim Hsu.

EXAMPLES:

sage: import sage.modular.arithgroup.arithgroup_perm as ap
sage: ap.HsuExample10()
Arithmetic subgroup with permutations of right cosets
S2=(1,2)(3,4)(5,6)(7,8)(9,10)
S3=(1,8,3)(2,4,6)(5,7,10)
L=(1,4)(2,5,9,10,8)(3,7,6)
R=(1,7,9,10,6)(2,3)(4,5,8)

sage.modular.arithgroup.arithgroup_perm.HsuExample18()

An example of an index 18 arithmetic subgroup studied by Tim Hsu.

EXAMPLES:

sage: import sage.modular.arithgroup.arithgroup_perm as ap
sage: ap.HsuExample18()
Arithmetic subgroup with permutations of right cosets
S2=(1,5)(2,11)(3,10)(4,15)(6,18)(7,12)(8,14)(9,16)(13,17)
S3=(1,7,11)(2,18,5)(3,9,15)(4,14,10)(6,17,12)(8,13,16)
L=(1,2)(3,4)(5,6,7)(8,9,10)(11,12,13,14,15,16,17,18)
R=(1,12,18)(2,6,13,9,4,8,17,7)(3,16,14)(5,11)(10,15)

class sage.modular.arithgroup.arithgroup_perm.OddArithmeticSubgroup_Permutation(S2, S3, L, R, canonical_labels=False)

An arithmetic subgroup of $$\SL(2, \ZZ)$$ not containing $$-1$$, represented in terms of the right action of $$\SL(2, \ZZ)$$ on its cosets.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2,3,4)",S3="(1,3)(2,4)")
sage: G
Arithmetic subgroup with permutations of right cosets
S2=(1,2,3,4)
S3=(1,3)(2,4)
L=(1,2,3,4)
R=(1,4,3,2)
sage: type(G)
<class 'sage.modular.arithgroup.arithgroup_perm.OddArithmeticSubgroup_Permutation_with_category'>

cusp_widths(exp=False)

Return the list of cusp widths.

INPUT:

exp - boolean (default: False) - if True, return a dictionary with keys the possible widths and with values the number of cusp with that width.

EXAMPLES:

sage: G = Gamma1(5).as_permutation_group()
sage: G.cusp_widths()
[1, 1, 5, 5]
sage: G.cusp_widths(exp=True)
{1: 2, 5: 2}

is_even()

Test whether the group is even.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,6,4,3)(2,7,5,8)",S3="(1,2,3,4,5,6)(7,8)")
sage: G.is_even()
False

is_odd()

Test whether the group is odd.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,6,4,3)(2,7,5,8)",S3="(1,2,3,4,5,6)(7,8)")
sage: G.is_odd()
True

ncusps()

Returns the number of cusps.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2,3,4)",S3="(1,3)(2,4)")
sage: G.ncusps()
1

sage: G = Gamma1(3).as_permutation_group()
sage: G.ncusps()
2

nirregcusps()

Return the number of irregular cusps.

The cusps are associated to cycles of the permutations $$L$$ or $$R$$. The irregular cusps are the one which are stabilised by $$-Id$$.

EXAMPLES:

sage: S2 = "(1,3,2,4)(5,7,6,8)(9,11,10,12)"
sage: S3 = "(1,3,5,2,4,6)(7,9,11,8,10,12)"
sage: G = ArithmeticSubgroup_Permutation(S2=S2,S3=S3)
sage: G.nirregcusps()
3

nregcusps()

Return the number of regular cusps of the group.

The cusps are associated to cycles of $$L$$ or $$R$$. The irregular cusps correspond to the ones which are not stabilised by $$-Id$$.

EXAMPLES:

sage: G = Gamma1(3).as_permutation_group()
sage: G.nregcusps()
2

nu2()

Return the number of elliptic points of order 2.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2,3,4)",S3="(1,3)(2,4)")
sage: G.nu2()
0

sage: G = Gamma1(2).as_permutation_group()
sage: G.nu2()
1

nu3()

Return the number of elliptic points of order 3.

EXAMPLES:

sage: G = ArithmeticSubgroup_Permutation(S2="(1,2,3,4)",S3="(1,3)(2,4)")
sage: G.nu3()
2

sage: G = Gamma1(3).as_permutation_group()
sage: G.nu3()
1

to_even_subgroup(relabel=True)

Returns the group with $$-Id$$ added in it.

EXAMPLES:

sage: G = Gamma1(3).as_permutation_group()
sage: G.to_even_subgroup()
Arithmetic subgroup with permutations of right cosets
S2=(1,3)(2,4)
S3=(1,2,3)
L=(2,3,4)
R=(1,4,2)

sage: H = ArithmeticSubgroup_Permutation(S2 = '(1,4,11,14)(2,7,12,17)(3,5,13,15)(6,9,16,19)(8,10,18,20)', S3 = '(1,2,3,11,12,13)(4,5,6,14,15,16)(7,8,9,17,18,19)(10,20)')
sage: G = H.to_even_subgroup(relabel=False); G
Arithmetic subgroup with permutations of right cosets
S2=(1,4)(2,7)(3,5)(6,9)(8,10)
S3=(1,2,3)(4,5,6)(7,8,9)
L=(1,5)(2,4,9,10,8)(3,7,6)
R=(1,7,10,8,6)(2,5,9)(3,4)
sage: H.is_subgroup(G)
True

sage.modular.arithgroup.arithgroup_perm.eval_sl2z_word(w)

Given a word in the format output by sl2z_word_problem(), convert it back into an element of $$\SL_2(\ZZ)$$.

EXAMPLES:

sage: from sage.modular.arithgroup.arithgroup_perm import eval_sl2z_word
sage: eval_sl2z_word([(0, 1), (1, -1), (0, 0), (1, 3), (0, 2), (1, 9), (0, -1)])
[ 66 -59]
[ 47 -42]

sage.modular.arithgroup.arithgroup_perm.sl2z_word_problem(A)

Given an element of $$\SL_2(\ZZ)$$, express it as a word in the generators L = [1,1,0,1] and R = [1,0,1,1].

The return format is a list of pairs (a,b), where a = 0 or 1 denoting L or R respectively, and b is an integer exponent.

See also the function eval_sl2z_word().

EXAMPLES:

sage: from sage.modular.arithgroup.arithgroup_perm import eval_sl2z_word, sl2z_word_problem
sage: m = SL2Z([1,0,0,1])
sage: eval_sl2z_word(sl2z_word_problem(m)) == m
True
sage: m = SL2Z([0,-1,1,0])
sage: eval_sl2z_word(sl2z_word_problem(m)) == m
True
sage: m = SL2Z([7,8,-50,-57])
sage: eval_sl2z_word(sl2z_word_problem(m)) == m
True

sage.modular.arithgroup.arithgroup_perm.word_of_perms(w, p1, p2)

Given a word $$w$$ as a list of 2-tuples (index,power) and permutations p1 and p2 return the product of p1 and p2 that corresponds to w.

EXAMPLES:

sage: import sage.modular.arithgroup.arithgroup_perm as ap
sage: S2 = SymmetricGroup(4)
sage: p1 = S2('(1,2)(3,4)')
sage: p2 = S2('(1,2,3,4)')
sage: ap.word_of_perms([(1,1),(0,1)], p1, p2) ==  p2 * p1
True
sage: ap.word_of_perms([(0,1),(1,1)], p1, p2) == p1 * p2
True