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39.3 The set $ \mathbf{P}^1(\mathbf{Q})$ of cusps

Module: sage.modular.cusps

The set $ \mathbf{P}^1(\mathbf{Q})$ 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 $ \mathbf{P}^1(\mathbf{Q})$, 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)
sage: loads(a.dumps()) == a
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 $ \Gamma_0(N)$ 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
sage: loads(C.dumps()) == C
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

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