The modular group \(\SL_2(\ZZ)\)#

AUTHORS:

Niles Johnson (2010-08): Issue #3893: random_element() should pass on *args and **kwds.

class sage.modular.arithgroup.congroup_sl2z.SL2Z_class[source]#

Bases: Gamma0_class

The full modular group \(\SL_2(\ZZ)\), regarded as a congruence subgroup of itself.

is_subgroup(right)[source]#

Return True if self is a subgroup of right.

EXAMPLES:

Sage

sage: SL2Z.is_subgroup(SL2Z)
True
sage: SL2Z.is_subgroup(Gamma1(1))
True
sage: SL2Z.is_subgroup(Gamma0(6))
False

Python

>>> from sage.all import *
>>> SL2Z.is_subgroup(SL2Z)
True
>>> SL2Z.is_subgroup(Gamma1(Integer(1)))
True
>>> SL2Z.is_subgroup(Gamma0(Integer(6)))
False

random_element(bound=100, *args, **kwds)[source]#

Return a random element of \(\SL_2(\ZZ)\) with entries whose absolute value is strictly less than bound (default 100). Additional arguments and keywords are passed to the random_element method of ZZ.

(Algorithm: Generate a random pair of integers at most bound. If they are not coprime, throw them away and start again. If they are, find an element of \(\SL_2(\ZZ)\) whose bottom row is that, and left-multiply it by \(\begin{pmatrix} 1 & w \\ 0 & 1\end{pmatrix}\) for an integer \(w\) randomly chosen from a small enough range that the answer still has entries at most bound.)

It is, unfortunately, not true that all elements of SL2Z with entries < bound appear with equal probability; those with larger bottom rows are favoured, because there are fewer valid possibilities for w.

EXAMPLES:

Sage

sage: s = SL2Z.random_element()
sage: s.parent() is SL2Z
True
sage: all(a in range(-99, 100) for a in list(s))
True
sage: S = set()
sage: while len(S) < 180:
....:     s = SL2Z.random_element(5)
....:     assert all(a in range(-4, 5) for a in list(s))
....:     assert s.parent() is SL2Z
....:     assert s in SL2Z
....:     S.add(s)

Python

>>> from sage.all import *
>>> s = SL2Z.random_element()
>>> s.parent() is SL2Z
True
>>> all(a in range(-Integer(99), Integer(100)) for a in list(s))
True
>>> S = set()
>>> while len(S) < Integer(180):
...     s = SL2Z.random_element(Integer(5))
...     assert all(a in range(-Integer(4), Integer(5)) for a in list(s))
...     assert s.parent() is SL2Z
...     assert s in SL2Z
...     S.add(s)

Passes extra positional or keyword arguments through:

Sage

sage: SL2Z.random_element(5, distribution='1/n').parent() is SL2Z
True

Python

>>> from sage.all import *
>>> SL2Z.random_element(Integer(5), distribution='1/n').parent() is SL2Z
True

reduce_cusp(c)[source]#

Return the unique reduced cusp equivalent to c under the action of self. Always returns Infinity, since there is only one equivalence class of cusps for \(SL_2(Z)\).

EXAMPLES:

Sage

sage: SL2Z.reduce_cusp(Cusps(-1/4))
Infinity

Python

>>> from sage.all import *
>>> SL2Z.reduce_cusp(Cusps(-Integer(1)/Integer(4)))
Infinity

sage.modular.arithgroup.congroup_sl2z.is_SL2Z(x)[source]#

Return True if x is the modular group \(\SL_2(\ZZ)\).

EXAMPLES:

Sage

sage: from sage.modular.arithgroup.all import is_SL2Z
sage: is_SL2Z(SL2Z)
doctest:warning...
DeprecationWarning: The function is_SL2Z is deprecated; use 'isinstance(..., SL2Z_class)' instead.
See https://github.com/sagemath/sage/issues/38035 for details.
True
sage: is_SL2Z(Gamma0(6))
False

Python

>>> from sage.all import *
>>> from sage.modular.arithgroup.all import is_SL2Z
>>> is_SL2Z(SL2Z)
doctest:warning...
DeprecationWarning: The function is_SL2Z is deprecated; use 'isinstance(..., SL2Z_class)' instead.
See https://github.com/sagemath/sage/issues/38035 for details.
True
>>> is_SL2Z(Gamma0(Integer(6)))
False