# 39.3 The set of cusps

Module: sage.modular.cusps

The set of cusps

sage: Cusps
Set P^1(QQ) of all cusps

sage: Cusp(oo)
Infinity

Class: Cusp

class Cusp
A cusp.

A cusp is either a rational number or infinity, i.e., an element of the projective line over Q. A Cusp is stored as a pair (a,b), where gcd(a,b)=1 and a,b are of type Integer.

sage: a = Cusp(2/3); b = Cusp(oo)
sage: a.parent()
Set P^1(QQ) of all cusps
sage: a.parent() is b.parent()
True
 Cusp( self, a, [b=1], [construct=False], [parent=None])
Create the cusp a/b in , where if b=0 this is the cusp at infinity.

sage: Cusp(2,3)
2/3
sage: Cusp(3,6)
1/2
sage: Cusp(1,0)
Infinity
sage: Cusp(infinity)
Infinity
sage: Cusp(5)
5
sage: Cusp(1/2)             # rational number
1/2
sage: Cusp(1.5)
3/2

sage: Cusp(sqrt(-1))
Traceback (most recent call last):
...
TypeError: unable to convert I to a rational

sage: a = Cusp(2,3)
True

Functions: apply, denominator, is_gamma0_equiv, is_gamma1_equiv, is_gamma_h_equiv, is_infinity, numerator

 apply( self, g)
Return g(self), where g=[a,b,c,d] is a list of length 4, which we view as a linear fractional transformation.

Apply the identity matrix:

sage: Cusp(0).apply([1,0,0,1])
0
sage: Cusp(0).apply([0,-1,1,0])
Infinity
sage: Cusp(0).apply([1,-3,0,1])
-3

 denominator( self)
Return the denominator of the cusp a/b.

sage: x=Cusp(6,9); x
2/3
sage: x.denominator()
3
sage: Cusp(oo).denominator()
0
sage: Cusp(-5/10).denominator()
2

 is_gamma0_equiv( self, other, N, [transformation=False])
Return whether self and other are equivalent modulo the action of via linear fractional transformations.

INPUT:
other -- Cusp
N -- an integer (specifies the group Gamma_0(N))
transformation -- bool (default: False), if True, also
return upper left entry of a matrix in
Gamma_0(N) that sends self to other.

OUTPUT:
bool -- True if self and other are equivalent
integer -- returned only if transformation is True

sage: x = Cusp(2,3)
sage: y = Cusp(4,5)
sage: x.is_gamma0_equiv(y, 2)
True
sage: x.is_gamma0_equiv(y, 2, True)
(True, 1)
sage: x.is_gamma0_equiv(y, 3)
False
sage: x.is_gamma0_equiv(y, 3, True)
(False, None)
sage: Cusp(1,0)
Infinity
sage: z = Cusp(1,0)
sage: x.is_gamma0_equiv(z, 3, True)
(True, 2)

ALGORITHM: See Proposition 2.2.3 of Cremona's book "Algorithms for Modular Elliptic Curves", or Prop 2.27 of Stein's Ph.D. thesis.

 is_gamma1_equiv( self, other, N)
Return whether self and other are equivalent modulo the action of Gamma_1(N) via linear fractional transformations.

INPUT:
other -- Cusp
N -- an integer (specifies the group Gamma_1(N))

OUTPUT:
bool -- True if self and other are equivalent
int -- 0, 1 or -1, gives further information
about the equivalence:  If the two cusps
are u1/v1 and u2/v2, then they are equivalent
if and only if
v1 = v2 (mod N) and u1 = u2 (mod gcd(v1,N))
or
v1 = -v2 (mod N) and u1 = -u2 (mod gcd(v1,N))
The sign is +1 for the first and -1 for the second.
If the two cusps are not equivalent then 0 is returned.

sage: x = Cusp(2,3)
sage: y = Cusp(4,5)
sage: x.is_gamma1_equiv(y,2)
(True, 1)
sage: x.is_gamma1_equiv(y,3)
(False, 0)
sage: z = Cusp(QQ(x) + 10)
sage: x.is_gamma1_equiv(z,10)
(True, 1)
sage: z = Cusp(1,0)
sage: x.is_gamma1_equiv(z, 3)
(True, -1)
sage: Cusp(0).is_gamma1_equiv(oo, 1)
(True, 1)
sage: Cusp(0).is_gamma1_equiv(oo, 3)
(False, 0)

 is_gamma_h_equiv( self, other, G)
Return whether self and other are equivalent modulo the action of Gamma_1(N) via linear fractional transformations.

INPUT:
other -- Cusp
G -- a congruence subgroup Gamma_H(N)

OUTPUT:
bool -- True if self and other are equivalent
int -- -1, 0, 1; extra info

sage: G = GammaH(13,[2])
sage: x = Cusp(2,3)
sage: y = Cusp(4,5)
sage: x.is_gamma_h_equiv(y,G)
(True, 1)

 is_infinity( self)
Returns True if this is the cusp infinity.

sage: Cusp(3/5).is_infinity()
False
sage: Cusp(1,0).is_infinity()
True
sage: Cusp(0,1).is_infinity()
False

 numerator( self)
Return the numerator of the cusp a/b.

sage: x=Cusp(6,9); x
2/3
sage: x.numerator()
2
sage: Cusp(oo).numerator()
1
sage: Cusp(-5/10).numerator()
-1

Special Functions: __cmp__, __neg__, _integer_, _latex_, _rational_, _repr_

 __cmp__( self, right)
Compare the cusps self and right. Comparison is as for rational numbers, except with the cusp oo greater than everything but itself.

The ordering in comparison is only really meaningful for infinity or elements that coerce to the rationals.

sage: Cusp(2/3) == Cusp(oo)
False

sage: Cusp(2/3) < Cusp(oo)
True

sage: Cusp(2/3)> Cusp(oo)
False

sage: Cusp(2/3) > Cusp(5/2)
False

sage: Cusp(2/3) < Cusp(5/2)
True

sage: Cusp(2/3) == Cusp(5/2)
False

sage: Cusp(oo) == Cusp(oo)
True

sage: 19/3 < Cusp(oo)
True

sage: Cusp(oo) < 19/3
False

sage: Cusp(2/3) < Cusp(11/7)
True

sage: Cusp(11/7) < Cusp(2/3)
False

sage: 2 < Cusp(3)
True

 __neg__( self)
The negative of this cusp.

sage: -Cusp(2/7)
-2/7
sage: -Cusp(oo)
Infinity

 _integer_( self)
Coerce to an integer.

sage: ZZ(Cusp(-19))
-19

sage: ZZ(Cusp(oo))
Traceback (most recent call last):
...
TypeError: cusp Infinity is not an integer
sage: ZZ(Cusp(-3,7))
Traceback (most recent call last):
...
TypeError: cusp -3/7 is not an integer

 _latex_( self)
Latex representation of this cusp.

sage: latex(Cusp(-2/7))
\frac{-2}{7}
sage: latex(Cusp(oo))
\infty

 _rational_( self)
Coerce to a rational number.

sage: QQ(Cusp(oo))
Traceback (most recent call last):
...
TypeError: cusp Infinity is not a rational number
sage: QQ(Cusp(-3,7))
-3/7

 _repr_( self)
String representation of this cusp.

sage: a = Cusp(2/3); a
2/3
sage: a.rename('2/3(cusp)'); a
2/3(cusp)

Class: Cusps_class

class Cusps_class
The set of cusps.

sage: C = Cusps; C
Set P^1(QQ) of all cusps
True
 Cusps_class( self)

Special Functions: __call__, __cmp__, _coerce_impl, _latex_, _repr_

 __call__( self, x)
Coerce x into the set of cusps.

sage: a = Cusps(-4/5); a
-4/5
sage: Cusps(a) is a
False
sage: Cusps(1.5)
3/2
sage: Cusps(oo)
Infinity
sage: Cusps(I)
Traceback (most recent call last):
...
TypeError: Unable to coerce I (<class 'sage.functions.constants.I_class'>)
to Rational

 __cmp__( self, right)
Return equality only if right is the set of cusps.

sage: Cusps == Cusps
True
sage: Cusps == QQ
False

 _coerce_impl( self, x)
Canonical coercion of x into the set of cusps.

sage: Cusps._coerce_(7/13)
7/13
sage: Cusps._coerce_(GF(7)(3))
Traceback (most recent call last):
...
TypeError: no canonical coercion of element into self
sage: Cusps(GF(7)(3))
3

 _latex_( self)
Return latex representation of self.

sage: latex(Cusps)
\mathbf{P}^1(\mathbf{Q})

 _repr_( self)
String representation of the set of cusps.

sage: Cusps
Set P^1(QQ) of all cusps
sage: Cusps.rename('CUSPS'); Cusps
CUSPS
sage: Cusps.rename(); Cusps
Set P^1(QQ) of all cusps
sage: Cusps
Set P^1(QQ) of all cusps