Package sage :: Package modular :: Module cusps :: Class Cusp
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Class Cusp


The set of cusps.

A cusp is either a rational number of infinity, i.e., an element of P1(Q). A Cusp is stored as a pair (a,b), where gcd(a,b)=1 and a,b are of type sage.rings.integers.Integer.
Method Summary
  __init__(self, a, b)
Create the cusp a/b in P^1(Q), where if b=0 it is the point at infinity.
  __neg__(self)
  __repr__(self)
  denominator(self)
Return the denominator of the cusp a/b.
  is_chi_equiv(self, other, chi)
  is_gamma0_equiv(self, other, N)
Return whether self and other are equivalent modulo Gamma_0(N).
  is_gamma1_equiv(self, other, N)
  numerator(self)
Return the numerator of the cusp a/b.

Method Details

__init__(self, a, b=1)
(Constructor)

Create the cusp a/b in P^1(Q), where if b=0 it is the point at infinity.
>>> Cusp(2,3)
2/3

>>> Cusp(3,6)
1/2

>>> Cusp(1,0)
Infinity

>>> Cusp(infinity)
Infinity

>>> Cusp(5)
5

>>> Cusp(Q("1/2"))             # rational number
1/2

>>> Cusp(1.5)
Traceback (most recent call last):

...

TypeError: Unable to coerce 1.5,1 to a Cusp

denominator(self)

Return the denominator of the cusp a/b.
>>> x=Cusp(6,9); x
2/3

>>> x.denominator()
3

is_gamma0_equiv(self, other, N)

Return whether self and other are equivalent modulo Gamma_0(N).

Keyword arguments:
    other -- a cusp
    N -- an integer (specifies the group Gamma_0(N))

Return True and an integer a if this cusp is equivalent to
other modulo the action of Gamma_0(N).  Here a is the upper
left entry of a matrix in Gamma_0(N) that sends self to other
(or is it other way around??).  If the two cusps self and
other are not congruent, then return (False, None).

numerator(self)

Return the numerator of the cusp a/b.
>>> x=Cusp(6,9); x
2/3

>>> x.numerator()
2

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