EXAMPLES:
sage: Cusps
Set P^1(QQ) of all cusps
sage: Cusp(oo)
Infinity
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.
EXAMPLES:
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
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.
EXAMPLES:
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
Create the cusp a/b in , where if b=0
this is the cusp at infinity.
When present, b must either be Infinity or coercible to an Integer.
EXAMPLES:
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)
1/2
sage: Cusp(1.5)
3/2
sage: Cusp(int(7))
7
sage: Cusp(1, 2, check=False)
1/2
sage: Cusp('sage', 2.5, check=False) # don't do this!
sage/2.50000000000000
sage: I**2
-1
sage: Cusp(I)
...
TypeError: Unable to convert I to a Cusp
sage: a = Cusp(2,3)
sage: loads(a.dumps()) == a
True
sage: Cusp(1/3,0)
Infinity
sage: Cusp((1,0))
Infinity
TESTS:
sage: Cusp("1/3", 5)
1/15
sage: Cusp(Cusp(3/5), 7)
3/35
sage: Cusp(5/3, 0)
Infinity
sage: Cusp(3,oo)
0
sage: Cusp((7,3), 5)
7/15
sage: Cusp(int(5), 7)
5/7
sage: Cusp(0,0)
...
TypeError: Unable to convert (0, 0) to a Cusp
sage: Cusp(oo,oo)
...
TypeError: Unable to convert (+Infinity, +Infinity) to a Cusp
sage: Cusp(Cusp(oo),oo)
...
TypeError: Unable to convert (Infinity, +Infinity) to a Cusp
The negative of this cusp.
EXAMPLES:
sage: -Cusp(2/7)
-2/7
sage: -Cusp(oo)
Infinity
Coerce to an integer.
EXAMPLES:
sage: ZZ(Cusp(-19))
-19
sage: Cusp(4,2)._integer_()
2
sage: ZZ(Cusp(oo))
...
TypeError: cusp Infinity is not an integer
sage: ZZ(Cusp(-3,7))
...
TypeError: cusp -3/7 is not an integer
Latex representation of this cusp.
EXAMPLES:
sage: latex(Cusp(-2/7))
\frac{-2}{7}
sage: latex(Cusp(oo))
\infty
sage: latex(Cusp(oo)) == Cusp(oo)._latex_()
True
Coerce to a rational number.
EXAMPLES:
sage: QQ(Cusp(oo))
...
TypeError: cusp Infinity is not a rational number
sage: QQ(Cusp(-3,7))
-3/7
sage: Cusp(11,2)._rational_()
11/2
String representation of this cusp.
EXAMPLES:
sage: a = Cusp(2/3); a
2/3
sage: a._repr_()
'2/3'
sage: a.rename('2/3(cusp)'); a
2/3(cusp)
Return g(self), where g=[a,b,c,d] is a list of length 4, which we view as a linear fractional transformation.
EXAMPLES: 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
Return the denominator of the cusp a/b.
EXAMPLES:
sage: x=Cusp(6,9); x
2/3
sage: x.denominator()
3
sage: Cusp(oo).denominator()
0
sage: Cusp(-5/10).denominator()
2
Suppose this cusp is ,
is a congruence subgroup of
level
, and
is the automorphism in the Galois
group of
that sends
to
. Then this function computes a cusp
such
that
, where
is the
equivalence class of
modulo
.
INPUT:
– integer that is coprime to N
– positive integer (level)
OUTPUT:
- a cusp
EXAMPLES:
sage: Cusp(1/10).galois_action(3, 50)
1/170
sage: Cusp(oo).galois_action(3, 50)
Infinity
sage: Cusp(0).galois_action(3, 50)
0
Here we compute explicitly the permutations of the action for t=3 on cusps for Gamma0(50):
sage: N = 50; t=3; G = Gamma0(N); C = G.cusps()
sage: cl = lambda z: exists(C, lambda y:y.is_gamma0_equiv(z, N))[1]
sage: for i in range(5): print i, t^i, [cl(alpha.galois_action(t^i,N)) for alpha in C]
0 1 [0, 1/25, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, Infinity]
1 3 [0, 1/25, 7/10, 2/5, 1/10, 4/5, 1/2, 1/5, 9/10, 3/5, 3/10, Infinity]
2 9 [0, 1/25, 9/10, 4/5, 7/10, 3/5, 1/2, 2/5, 3/10, 1/5, 1/10, Infinity]
3 27 [0, 1/25, 3/10, 3/5, 9/10, 1/5, 1/2, 4/5, 1/10, 2/5, 7/10, Infinity]
4 81 [0, 1/25, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, Infinity]
REFERENCES:
- Section 1.3 of Glenn Stevens, “Arithmetic on Modular Curves”
- There is a long comment about our algorithm in the source code for this function.
AUTHORS:
- William Stein, 2009-04-18
Return whether self and other are equivalent modulo the action of
via linear fractional transformations.
INPUT:
OUTPUT:
EXAMPLES:
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.
Return whether self and other are equivalent modulo the action of Gamma_1(N) via linear fractional transformations.
INPUT:
OUTPUT:
EXAMPLES:
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)
Return a pair (b, t), where b is True or False as self and other are equivalent under the action of G, and t is 1 or -1, as described below.
Two cusps and
are equivalent modulo
Gamma_H(N) if and only if
and
or
and
for some
. Then t is 1 or -1 as c and c’ fall into the
first or second case, respectively.
INPUT:
OUTPUT:
EXAMPLES:
sage: x = Cusp(2,3)
sage: y = Cusp(4,5)
sage: x.is_gamma_h_equiv(y,GammaH(13,[2]))
(True, 1)
sage: x.is_gamma_h_equiv(y,GammaH(13,[5]))
(False, 0)
sage: x.is_gamma_h_equiv(y,GammaH(5,[]))
(False, 0)
sage: x.is_gamma_h_equiv(y,GammaH(23,[4]))
(True, -1)
Enumerating the cusps for a space of modular symbols uses this function.
sage: G = GammaH(25,[6]) ; M = G.modular_symbols() ; M
Modular Symbols space of dimension 11 for Congruence Subgroup Gamma_H(25) with H generated by [6] of weight 2 with sign 0 and over Rational Field
sage: M.cusps()
[37/75, 1/2, 31/125, 1/4, -2/5, 2/5, -1/5, 1/10, -3/10, 1/15, 7/15, 9/20]
sage: len(M.cusps())
12
This is always one more than the associated space of weight 2 Eisenstein series.
sage: G.dimension_eis(2)
11
sage: M.cuspidal_subspace()
Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 11 for Congruence Subgroup Gamma_H(25) with H generated by [6] of weight 2 with sign 0 and over Rational Field
sage: G.dimension_cusp_forms(2)
0
Returns True if this is the cusp infinity.
EXAMPLES:
sage: Cusp(3/5).is_infinity()
False
sage: Cusp(1,0).is_infinity()
True
sage: Cusp(0,1).is_infinity()
False
Return the numerator of the cusp a/b.
EXAMPLES:
sage: x=Cusp(6,9); x
2/3
sage: x.numerator()
2
sage: Cusp(oo).numerator()
1
sage: Cusp(-5/10).numerator()
-1
The set of cusps.
EXAMPLES:
sage: C = Cusps; C
Set P^1(QQ) of all cusps
sage: loads(C.dumps()) == C
True
Coerce x into the set of cusps.
EXAMPLES:
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)
...
TypeError: Unable to convert I to a Cusp
Return equality only if right is the set of cusps.
EXAMPLES:
sage: Cusps == Cusps
True
sage: Cusps == QQ
False
The set of cusps, i.e. .
EXAMPLES:
sage: C = sage.modular.cusps.Cusps_class() ; C
Set P^1(QQ) of all cusps
sage: Cusps == C
True
Canonical coercion of x into the set of cusps.
EXAMPLES:
sage: Cusps._coerce_(7/13)
7/13
sage: Cusps._coerce_(GF(7)(3))
...
TypeError: no canonical coercion of element into self
sage: Cusps(GF(7)(3))
3
sage: Cusps._coerce_impl(GF(7)(3))
...
TypeError: no canonical coercion of element into self
Return latex representation of self.
EXAMPLES:
sage: latex(Cusps)
\mathbf{P}^1(\QQ)
sage: latex(Cusps) == Cusps._latex_()
True
String representation of the set of cusps.
EXAMPLES:
sage: Cusps
Set P^1(QQ) of all cusps
sage: Cusps._repr_()
'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