The set \(\mathbb{P}^1(K)\) of cusps of a number field \(K\)¶

AUTHORS:

Maite Aranes (2009): Initial version

EXAMPLES:

The space of cusps over a number field k:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 5)
sage: kCusps = NFCusps(k); kCusps
Set of all cusps of Number Field in a with defining polynomial x^2 + 5
sage: kCusps is NFCusps(k)
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(5), names=('a',)); (a,) = k._first_ngens(1)
>>> kCusps = NFCusps(k); kCusps
Set of all cusps of Number Field in a with defining polynomial x^2 + 5
>>> kCusps is NFCusps(k)
True

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 5)
kCusps = NFCusps(k); kCusps
kCusps is NFCusps(k)

Define a cusp over a number field:

Sage

sage: NFCusp(k, a, 2/(a+1))
Cusp [a - 5: 2] of Number Field in a with defining polynomial x^2 + 5
sage: kCusps((a,2))
Cusp [a: 2] of Number Field in a with defining polynomial x^2 + 5
sage: NFCusp(k,oo)
Cusp Infinity of Number Field in a with defining polynomial x^2 + 5

Python

>>> from sage.all import *
>>> NFCusp(k, a, Integer(2)/(a+Integer(1)))
Cusp [a - 5: 2] of Number Field in a with defining polynomial x^2 + 5
>>> kCusps((a,Integer(2)))
Cusp [a: 2] of Number Field in a with defining polynomial x^2 + 5
>>> NFCusp(k,oo)
Cusp Infinity of Number Field in a with defining polynomial x^2 + 5

Sage Live

NFCusp(k, a, 2/(a+1))
kCusps((a,2))
NFCusp(k,oo)

Different operations with cusps over a number field:

Sage

sage: alpha = NFCusp(k, 3, 1/a + 2); alpha
Cusp [a + 10: 7] of Number Field in a with defining polynomial x^2 + 5
sage: alpha.numerator()
a + 10
sage: alpha.denominator()
7
sage: alpha.ideal()
Fractional ideal (7, a + 3)
sage: M = alpha.ABmatrix(); M # random
[a + 10, 2*a + 6, 7, a + 5]
sage: NFCusp(k, oo).apply(M)
Cusp [a + 10: 7] of Number Field in a with defining polynomial x^2 + 5

Python

>>> from sage.all import *
>>> alpha = NFCusp(k, Integer(3), Integer(1)/a + Integer(2)); alpha
Cusp [a + 10: 7] of Number Field in a with defining polynomial x^2 + 5
>>> alpha.numerator()
a + 10
>>> alpha.denominator()
7
>>> alpha.ideal()
Fractional ideal (7, a + 3)
>>> M = alpha.ABmatrix(); M # random
[a + 10, 2*a + 6, 7, a + 5]
>>> NFCusp(k, oo).apply(M)
Cusp [a + 10: 7] of Number Field in a with defining polynomial x^2 + 5

Sage Live

alpha = NFCusp(k, 3, 1/a + 2); alpha
alpha.numerator()
alpha.denominator()
alpha.ideal()
M = alpha.ABmatrix(); M # random
NFCusp(k, oo).apply(M)

Check Gamma0(N)-equivalence of cusps:

Sage

sage: N = k.ideal(3)
sage: alpha = NFCusp(k, 3, a + 1)
sage: beta = kCusps((2, a - 3))
sage: alpha.is_Gamma0_equivalent(beta, N)
True

Python

>>> from sage.all import *
>>> N = k.ideal(Integer(3))
>>> alpha = NFCusp(k, Integer(3), a + Integer(1))
>>> beta = kCusps((Integer(2), a - Integer(3)))
>>> alpha.is_Gamma0_equivalent(beta, N)
True

Sage Live

N = k.ideal(3)
alpha = NFCusp(k, 3, a + 1)
beta = kCusps((2, a - 3))
alpha.is_Gamma0_equivalent(beta, N)

Obtain transformation matrix for equivalent cusps:

Sage

sage: t, M = alpha.is_Gamma0_equivalent(beta, N, Transformation=True)
sage: M[2] in N
True
sage: M[0]*M[3] - M[1]*M[2] == 1
True
sage: alpha.apply(M) == beta
True

Python

>>> from sage.all import *
>>> t, M = alpha.is_Gamma0_equivalent(beta, N, Transformation=True)
>>> M[Integer(2)] in N
True
>>> M[Integer(0)]*M[Integer(3)] - M[Integer(1)]*M[Integer(2)] == Integer(1)
True
>>> alpha.apply(M) == beta
True

Sage Live

t, M = alpha.is_Gamma0_equivalent(beta, N, Transformation=True)
M[2] in N
M[0]*M[3] - M[1]*M[2] == 1
alpha.apply(M) == beta

List representatives for Gamma_0(N) - equivalence classes of cusps:

Sage

sage: Gamma0_NFCusps(N)
[Cusp [0: 1] of Number Field in a with defining polynomial x^2 + 5,
 Cusp [1: 3] of Number Field in a with defining polynomial x^2 + 5,
...]

Python

>>> from sage.all import *
>>> Gamma0_NFCusps(N)
[Cusp [0: 1] of Number Field in a with defining polynomial x^2 + 5,
 Cusp [1: 3] of Number Field in a with defining polynomial x^2 + 5,
...]

Sage Live

Gamma0_NFCusps(N)

sage.modular.cusps_nf.Gamma0_NFCusps(N)[source]¶

Return a list of inequivalent cusps for \(\Gamma_0(N)\), i.e., a set of representatives for the orbits of self on \(\mathbb{P}^1(k)\).

INPUT:

N – an integral ideal of the number field k (the level)

OUTPUT: list of inequivalent number field cusps

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 5)
sage: N = k.ideal(3)
sage: L = Gamma0_NFCusps(N)

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(5), names=('a',)); (a,) = k._first_ngens(1)
>>> N = k.ideal(Integer(3))
>>> L = Gamma0_NFCusps(N)

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 5)
N = k.ideal(3)
L = Gamma0_NFCusps(N)

The cusps in the list are inequivalent:

Sage

sage: any(L[i].is_Gamma0_equivalent(L[j], N)
....:     for i in range(len(L)) for j in range(len(L)) if i < j)
False

Python

>>> from sage.all import *
>>> any(L[i].is_Gamma0_equivalent(L[j], N)
...     for i in range(len(L)) for j in range(len(L)) if i < j)
False

Sage Live

any(L[i].is_Gamma0_equivalent(L[j], N)
    for i in range(len(L)) for j in range(len(L)) if i < j)

We test that we obtain the right number of orbits:

Sage

sage: from sage.modular.cusps_nf import number_of_Gamma0_NFCusps
sage: len(L) == number_of_Gamma0_NFCusps(N)
True

Python

>>> from sage.all import *
>>> from sage.modular.cusps_nf import number_of_Gamma0_NFCusps
>>> len(L) == number_of_Gamma0_NFCusps(N)
True

Sage Live

from sage.modular.cusps_nf import number_of_Gamma0_NFCusps
len(L) == number_of_Gamma0_NFCusps(N)

Another example:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^4 - x^3 -21*x^2 + 17*x + 133)
sage: N = k.ideal(5)
sage: from sage.modular.cusps_nf import number_of_Gamma0_NFCusps
sage: len(Gamma0_NFCusps(N)) == number_of_Gamma0_NFCusps(N) # long time (over 1 sec)
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(4) - x**Integer(3) -Integer(21)*x**Integer(2) + Integer(17)*x + Integer(133), names=('a',)); (a,) = k._first_ngens(1)
>>> N = k.ideal(Integer(5))
>>> from sage.modular.cusps_nf import number_of_Gamma0_NFCusps
>>> len(Gamma0_NFCusps(N)) == number_of_Gamma0_NFCusps(N) # long time (over 1 sec)
True

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^4 - x^3 -21*x^2 + 17*x + 133)
N = k.ideal(5)
from sage.modular.cusps_nf import number_of_Gamma0_NFCusps
len(Gamma0_NFCusps(N)) == number_of_Gamma0_NFCusps(N) # long time (over 1 sec)

class sage.modular.cusps_nf.NFCusp(number_field, a, b=None, parent=None, lreps=None)[source]¶

Bases: Element

Create a number field cusp, i.e., an element of \(\mathbb{P}^1(k)\).

A cusp on a number field is either an element of the field or infinity, i.e., an element of the projective line over the number field. It is stored as a pair (a,b), where a, b are integral elements of the number field.

INPUT:

number_field – the number field over which the cusp is defined
a – it can be a number field element (integral or not), or a number field cusp
b – (optional) when present, it must be either Infinity or coercible to an element of the number field
lreps – (optional) a list of chosen representatives for all the ideal classes of the field. When given, the representative of the cusp will be changed so its associated ideal is one of the ideals in the list.

OUTPUT:

[a: b] – a number field cusp.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 5)
sage: NFCusp(k, a, 2)
Cusp [a: 2] of Number Field in a with defining polynomial x^2 + 5
sage: NFCusp(k, (a,2))
Cusp [a: 2] of Number Field in a with defining polynomial x^2 + 5
sage: NFCusp(k, a, 2/(a+1))
Cusp [a - 5: 2] of Number Field in a with defining polynomial x^2 + 5

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(5), names=('a',)); (a,) = k._first_ngens(1)
>>> NFCusp(k, a, Integer(2))
Cusp [a: 2] of Number Field in a with defining polynomial x^2 + 5
>>> NFCusp(k, (a,Integer(2)))
Cusp [a: 2] of Number Field in a with defining polynomial x^2 + 5
>>> NFCusp(k, a, Integer(2)/(a+Integer(1)))
Cusp [a - 5: 2] of Number Field in a with defining polynomial x^2 + 5

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 5)
NFCusp(k, a, 2)
NFCusp(k, (a,2))
NFCusp(k, a, 2/(a+1))

Cusp Infinity:

Sage

sage: NFCusp(k, 0)
Cusp [0: 1] of Number Field in a with defining polynomial x^2 + 5
sage: NFCusp(k, oo)
Cusp Infinity of Number Field in a with defining polynomial x^2 + 5
sage: NFCusp(k, 3*a, oo)
Cusp [0: 1] of Number Field in a with defining polynomial x^2 + 5
sage: NFCusp(k, a + 5, 0)
Cusp Infinity of Number Field in a with defining polynomial x^2 + 5

Python

>>> from sage.all import *
>>> NFCusp(k, Integer(0))
Cusp [0: 1] of Number Field in a with defining polynomial x^2 + 5
>>> NFCusp(k, oo)
Cusp Infinity of Number Field in a with defining polynomial x^2 + 5
>>> NFCusp(k, Integer(3)*a, oo)
Cusp [0: 1] of Number Field in a with defining polynomial x^2 + 5
>>> NFCusp(k, a + Integer(5), Integer(0))
Cusp Infinity of Number Field in a with defining polynomial x^2 + 5

Sage Live

NFCusp(k, 0)
NFCusp(k, oo)
NFCusp(k, 3*a, oo)
NFCusp(k, a + 5, 0)

Saving and loading works:

Sage

sage: alpha = NFCusp(k, a, 2/(a+1))
sage: loads(dumps(alpha))==alpha
True

Python

>>> from sage.all import *
>>> alpha = NFCusp(k, a, Integer(2)/(a+Integer(1)))
>>> loads(dumps(alpha))==alpha
True

Sage Live

alpha = NFCusp(k, a, 2/(a+1))
loads(dumps(alpha))==alpha

Some tests:

Sage

sage: I*I
-1
sage: NFCusp(k, I)
Traceback (most recent call last):
...
TypeError: unable to convert I to a cusp of the number field

Python

>>> from sage.all import *
>>> I*I
-1
>>> NFCusp(k, I)
Traceback (most recent call last):
...
TypeError: unable to convert I to a cusp of the number field

Sage Live

I*I
NFCusp(k, I)

Sage

sage: NFCusp(k, oo, oo)
Traceback (most recent call last):
...
TypeError: unable to convert (+Infinity, +Infinity) to a cusp of the number field

Python

>>> from sage.all import *
>>> NFCusp(k, oo, oo)
Traceback (most recent call last):
...
TypeError: unable to convert (+Infinity, +Infinity) to a cusp of the number field

Sage Live

NFCusp(k, oo, oo)

Python

>>> from sage.all import *
>>> NFCusp(k, oo, oo)
Traceback (most recent call last):
...
TypeError: unable to convert (+Infinity, +Infinity) to a cusp of the number field

Sage Live

NFCusp(k, oo, oo)

Sage

sage: NFCusp(k, 0, 0)
Traceback (most recent call last):
...
TypeError: unable to convert (0, 0) to a cusp of the number field

Python

>>> from sage.all import *
>>> NFCusp(k, Integer(0), Integer(0))
Traceback (most recent call last):
...
TypeError: unable to convert (0, 0) to a cusp of the number field

Sage Live

NFCusp(k, 0, 0)

Python

>>> from sage.all import *
>>> NFCusp(k, Integer(0), Integer(0))
Traceback (most recent call last):
...
TypeError: unable to convert (0, 0) to a cusp of the number field

Sage Live

NFCusp(k, 0, 0)

Sage

sage: NFCusp(k, "a + 2", a)
Cusp [-2*a + 5: 5] of Number Field in a with defining polynomial x^2 + 5

Python

>>> from sage.all import *
>>> NFCusp(k, "a + 2", a)
Cusp [-2*a + 5: 5] of Number Field in a with defining polynomial x^2 + 5

Sage Live

NFCusp(k, "a + 2", a)

Python

>>> from sage.all import *
>>> NFCusp(k, "a + 2", a)
Cusp [-2*a + 5: 5] of Number Field in a with defining polynomial x^2 + 5

Sage Live

NFCusp(k, "a + 2", a)

Sage

sage: NFCusp(k, NFCusp(k, oo))
Cusp Infinity of Number Field in a with defining polynomial x^2 + 5
sage: c = NFCusp(k, 3, 2*a)
sage: NFCusp(k, c, a + 1)
Cusp [-a - 5: 20] of Number Field in a with defining polynomial x^2 + 5
sage: L.<b> = NumberField(x^2 + 2)
sage: NFCusp(L, c)
Traceback (most recent call last):
...
ValueError: Cannot coerce cusps from one field to another

Python

>>> from sage.all import *
>>> NFCusp(k, NFCusp(k, oo))
Cusp Infinity of Number Field in a with defining polynomial x^2 + 5
>>> c = NFCusp(k, Integer(3), Integer(2)*a)
>>> NFCusp(k, c, a + Integer(1))
Cusp [-a - 5: 20] of Number Field in a with defining polynomial x^2 + 5
>>> L = NumberField(x**Integer(2) + Integer(2), names=('b',)); (b,) = L._first_ngens(1)
>>> NFCusp(L, c)
Traceback (most recent call last):
...
ValueError: Cannot coerce cusps from one field to another

Sage Live

NFCusp(k, NFCusp(k, oo))
c = NFCusp(k, 3, 2*a)
NFCusp(k, c, a + 1)
L.<b> = NumberField(x^2 + 2)
NFCusp(L, c)

Python

>>> from sage.all import *
>>> NFCusp(k, NFCusp(k, oo))
Cusp Infinity of Number Field in a with defining polynomial x^2 + 5
>>> c = NFCusp(k, Integer(3), Integer(2)*a)
>>> NFCusp(k, c, a + Integer(1))
Cusp [-a - 5: 20] of Number Field in a with defining polynomial x^2 + 5
>>> L = NumberField(x**Integer(2) + Integer(2), names=('b',)); (b,) = L._first_ngens(1)
>>> NFCusp(L, c)
Traceback (most recent call last):
...
ValueError: Cannot coerce cusps from one field to another

Sage Live

NFCusp(k, NFCusp(k, oo))
c = NFCusp(k, 3, 2*a)
NFCusp(k, c, a + 1)
L.<b> = NumberField(x^2 + 2)
NFCusp(L, c)

ABmatrix()[source]¶

Return AB-matrix associated to the cusp self.

Given R a Dedekind domain and A, B ideals of R in inverse classes, an AB-matrix is a matrix realizing the isomorphism between R+R and A+B. An AB-matrix associated to a cusp [a1: a2] is an AB-matrix with A the ideal associated to the cusp (A=<a1, a2>) and first column given by the coefficients of the cusp.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^3 + 11)
sage: alpha = NFCusp(k, oo)
sage: alpha.ABmatrix()
[1, 0, 0, 1]

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(3) + Integer(11), names=('a',)); (a,) = k._first_ngens(1)
>>> alpha = NFCusp(k, oo)
>>> alpha.ABmatrix()
[1, 0, 0, 1]

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^3 + 11)
alpha = NFCusp(k, oo)
alpha.ABmatrix()

Sage

sage: alpha = NFCusp(k, 0)
sage: alpha.ABmatrix()
[0, -1, 1, 0]

Python

>>> from sage.all import *
>>> alpha = NFCusp(k, Integer(0))
>>> alpha.ABmatrix()
[0, -1, 1, 0]

Sage Live

alpha = NFCusp(k, 0)
alpha.ABmatrix()

Python

>>> from sage.all import *
>>> alpha = NFCusp(k, Integer(0))
>>> alpha.ABmatrix()
[0, -1, 1, 0]

Sage Live

alpha = NFCusp(k, 0)
alpha.ABmatrix()

Note that the AB-matrix associated to a cusp is not unique, and the output of the ABmatrix function may change.

Sage

sage: alpha = NFCusp(k, 3/2, a-1)
sage: M = alpha.ABmatrix()
sage: M # random
[-a^2 - a - 1, -3*a - 7, 8, -2*a^2 - 3*a + 4]
sage: M[0] == alpha.numerator() and M[2] == alpha.denominator()
True

Python

>>> from sage.all import *
>>> alpha = NFCusp(k, Integer(3)/Integer(2), a-Integer(1))
>>> M = alpha.ABmatrix()
>>> M # random
[-a^2 - a - 1, -3*a - 7, 8, -2*a^2 - 3*a + 4]
>>> M[Integer(0)] == alpha.numerator() and M[Integer(2)] == alpha.denominator()
True

Sage Live

alpha = NFCusp(k, 3/2, a-1)
M = alpha.ABmatrix()
M # random
M[0] == alpha.numerator() and M[2] == alpha.denominator()

An AB-matrix associated to a cusp alpha will send Infinity to alpha:

Sage

sage: alpha = NFCusp(k, 3, a-1)
sage: M = alpha.ABmatrix()
sage: (k.ideal(M[1], M[3])*alpha.ideal()).is_principal()
True
sage: M[0] == alpha.numerator() and M[2] == alpha.denominator()
True
sage: NFCusp(k, oo).apply(M) == alpha
True

Python

>>> from sage.all import *
>>> alpha = NFCusp(k, Integer(3), a-Integer(1))
>>> M = alpha.ABmatrix()
>>> (k.ideal(M[Integer(1)], M[Integer(3)])*alpha.ideal()).is_principal()
True
>>> M[Integer(0)] == alpha.numerator() and M[Integer(2)] == alpha.denominator()
True
>>> NFCusp(k, oo).apply(M) == alpha
True

Sage Live

alpha = NFCusp(k, 3, a-1)
M = alpha.ABmatrix()
(k.ideal(M[1], M[3])*alpha.ideal()).is_principal()
M[0] == alpha.numerator() and M[2] == alpha.denominator()
NFCusp(k, oo).apply(M) == alpha

apply(g)[source]¶

Return g(self), where g is a 2x2 matrix, which we view as a linear fractional transformation.

INPUT:

g – list of integral elements [a, b, c, d] that are the entries of a 2x2 matrix

OUTPUT:

A number field cusp, obtained by the action of g on the cusp self.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 23)
sage: beta = NFCusp(k, 0, 1)
sage: beta.apply([0, -1, 1, 0])
Cusp Infinity of Number Field in a with defining polynomial x^2 + 23
sage: beta.apply([1, a, 0, 1])
Cusp [a: 1] of Number Field in a with defining polynomial x^2 + 23

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = k._first_ngens(1)
>>> beta = NFCusp(k, Integer(0), Integer(1))
>>> beta.apply([Integer(0), -Integer(1), Integer(1), Integer(0)])
Cusp Infinity of Number Field in a with defining polynomial x^2 + 23
>>> beta.apply([Integer(1), a, Integer(0), Integer(1)])
Cusp [a: 1] of Number Field in a with defining polynomial x^2 + 23

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 23)
beta = NFCusp(k, 0, 1)
beta.apply([0, -1, 1, 0])
beta.apply([1, a, 0, 1])

denominator()[source]¶

Return the denominator of the cusp self.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 1)
sage: c = NFCusp(k, a, 2)
sage: c.denominator()
2
sage: d = NFCusp(k, 1, a + 1);d
Cusp [1: a + 1] of Number Field in a with defining polynomial x^2 + 1
sage: d.denominator()
a + 1
sage: NFCusp(k, oo).denominator()
0

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = k._first_ngens(1)
>>> c = NFCusp(k, a, Integer(2))
>>> c.denominator()
2
>>> d = NFCusp(k, Integer(1), a + Integer(1));d
Cusp [1: a + 1] of Number Field in a with defining polynomial x^2 + 1
>>> d.denominator()
a + 1
>>> NFCusp(k, oo).denominator()
0

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 1)
c = NFCusp(k, a, 2)
c.denominator()
d = NFCusp(k, 1, a + 1);d
d.denominator()
NFCusp(k, oo).denominator()

ideal()[source]¶

Return the ideal associated to the cusp self.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 23)
sage: alpha = NFCusp(k, 3, a-1)
sage: alpha.ideal()
Fractional ideal (3, 1/2*a - 1/2)
sage: NFCusp(k, oo).ideal()
Fractional ideal (1)

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = k._first_ngens(1)
>>> alpha = NFCusp(k, Integer(3), a-Integer(1))
>>> alpha.ideal()
Fractional ideal (3, 1/2*a - 1/2)
>>> NFCusp(k, oo).ideal()
Fractional ideal (1)

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 23)
alpha = NFCusp(k, 3, a-1)
alpha.ideal()
NFCusp(k, oo).ideal()

is_Gamma0_equivalent(other, N, Transformation=False)[source]¶

Check if cusps self and other are \(\Gamma_0(N)\)- equivalent.

INPUT:

other – a number field cusp or a list of two number field elements which define a cusp
N – an ideal of the number field (level)

OUTPUT: boolean; True if the cusps are equivalent

a transformation matrix – (if Transformation=True) a list of integral elements [a, b, c, d] which are the entries of a 2x2 matrix M in \(\Gamma_0(N)\) such that M * self = other if other and self are \(\Gamma_0(N)\)- equivalent. If self and other are not equivalent it returns zero.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^3 - 10)
sage: N = K.ideal(a - 1)
sage: alpha = NFCusp(K, 0)
sage: beta = NFCusp(K, oo)
sage: alpha.is_Gamma0_equivalent(beta, N)
False
sage: alpha.is_Gamma0_equivalent(beta, K.ideal(1))
True
sage: b, M = alpha.is_Gamma0_equivalent(beta, K.ideal(1),Transformation=True)
sage: alpha.apply(M)
Cusp Infinity of Number Field in a with defining polynomial x^3 - 10

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(10), names=('a',)); (a,) = K._first_ngens(1)
>>> N = K.ideal(a - Integer(1))
>>> alpha = NFCusp(K, Integer(0))
>>> beta = NFCusp(K, oo)
>>> alpha.is_Gamma0_equivalent(beta, N)
False
>>> alpha.is_Gamma0_equivalent(beta, K.ideal(Integer(1)))
True
>>> b, M = alpha.is_Gamma0_equivalent(beta, K.ideal(Integer(1)),Transformation=True)
>>> alpha.apply(M)
Cusp Infinity of Number Field in a with defining polynomial x^3 - 10

Sage Live

x = polygen(ZZ, 'x')
K.<a> = NumberField(x^3 - 10)
N = K.ideal(a - 1)
alpha = NFCusp(K, 0)
beta = NFCusp(K, oo)
alpha.is_Gamma0_equivalent(beta, N)
alpha.is_Gamma0_equivalent(beta, K.ideal(1))
b, M = alpha.is_Gamma0_equivalent(beta, K.ideal(1),Transformation=True)
alpha.apply(M)

Sage

sage: k.<a> = NumberField(x^2 + 23)
sage: N = k.ideal(3)
sage: alpha1 = NFCusp(k, a+1, 4)
sage: alpha2 = NFCusp(k, a-8, 29)
sage: alpha1.is_Gamma0_equivalent(alpha2, N)
True
sage: b, M = alpha1.is_Gamma0_equivalent(alpha2, N, Transformation=True)
sage: alpha1.apply(M) == alpha2
True
sage: M[2] in N
True

Python

>>> from sage.all import *
>>> k = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = k._first_ngens(1)
>>> N = k.ideal(Integer(3))
>>> alpha1 = NFCusp(k, a+Integer(1), Integer(4))
>>> alpha2 = NFCusp(k, a-Integer(8), Integer(29))
>>> alpha1.is_Gamma0_equivalent(alpha2, N)
True
>>> b, M = alpha1.is_Gamma0_equivalent(alpha2, N, Transformation=True)
>>> alpha1.apply(M) == alpha2
True
>>> M[Integer(2)] in N
True

Sage Live

k.<a> = NumberField(x^2 + 23)
N = k.ideal(3)
alpha1 = NFCusp(k, a+1, 4)
alpha2 = NFCusp(k, a-8, 29)
alpha1.is_Gamma0_equivalent(alpha2, N)
b, M = alpha1.is_Gamma0_equivalent(alpha2, N, Transformation=True)
alpha1.apply(M) == alpha2
M[2] in N

Python

>>> from sage.all import *
>>> k = NumberField(x**Integer(2) + Integer(23), names=('a',)); (a,) = k._first_ngens(1)
>>> N = k.ideal(Integer(3))
>>> alpha1 = NFCusp(k, a+Integer(1), Integer(4))
>>> alpha2 = NFCusp(k, a-Integer(8), Integer(29))
>>> alpha1.is_Gamma0_equivalent(alpha2, N)
True
>>> b, M = alpha1.is_Gamma0_equivalent(alpha2, N, Transformation=True)
>>> alpha1.apply(M) == alpha2
True
>>> M[Integer(2)] in N
True

Sage Live

k.<a> = NumberField(x^2 + 23)
N = k.ideal(3)
alpha1 = NFCusp(k, a+1, 4)
alpha2 = NFCusp(k, a-8, 29)
alpha1.is_Gamma0_equivalent(alpha2, N)
b, M = alpha1.is_Gamma0_equivalent(alpha2, N, Transformation=True)
alpha1.apply(M) == alpha2
M[2] in N

is_infinity()[source]¶

Return True if this is the cusp infinity.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 1)
sage: NFCusp(k, a, 2).is_infinity()
False
sage: NFCusp(k, 2, 0).is_infinity()
True
sage: NFCusp(k, oo).is_infinity()
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = k._first_ngens(1)
>>> NFCusp(k, a, Integer(2)).is_infinity()
False
>>> NFCusp(k, Integer(2), Integer(0)).is_infinity()
True
>>> NFCusp(k, oo).is_infinity()
True

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 1)
NFCusp(k, a, 2).is_infinity()
NFCusp(k, 2, 0).is_infinity()
NFCusp(k, oo).is_infinity()

number_field()[source]¶

Return the number field of definition of the cusp self.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 2)
sage: alpha = NFCusp(k, 1, a + 1)
sage: alpha.number_field()
Number Field in a with defining polynomial x^2 + 2

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(2), names=('a',)); (a,) = k._first_ngens(1)
>>> alpha = NFCusp(k, Integer(1), a + Integer(1))
>>> alpha.number_field()
Number Field in a with defining polynomial x^2 + 2

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 2)
alpha = NFCusp(k, 1, a + 1)
alpha.number_field()

numerator()[source]¶

Return the numerator of the cusp self.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 1)
sage: c = NFCusp(k, a, 2)
sage: c.numerator()
a
sage: d = NFCusp(k, 1, a)
sage: d.numerator()
1
sage: NFCusp(k, oo).numerator()
1

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = k._first_ngens(1)
>>> c = NFCusp(k, a, Integer(2))
>>> c.numerator()
a
>>> d = NFCusp(k, Integer(1), a)
>>> d.numerator()
1
>>> NFCusp(k, oo).numerator()
1

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 1)
c = NFCusp(k, a, 2)
c.numerator()
d = NFCusp(k, 1, a)
d.numerator()
NFCusp(k, oo).numerator()

sage.modular.cusps_nf.NFCusps()[source]¶

The set of cusps of a number field \(K\), i.e. \(\mathbb{P}^1(K)\).

INPUT:

number_field – a number field

OUTPUT: the set of cusps over the given number field

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 5)
sage: kCusps = NFCusps(k); kCusps
Set of all cusps of Number Field in a with defining polynomial x^2 + 5
sage: kCusps is NFCusps(k)
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(5), names=('a',)); (a,) = k._first_ngens(1)
>>> kCusps = NFCusps(k); kCusps
Set of all cusps of Number Field in a with defining polynomial x^2 + 5
>>> kCusps is NFCusps(k)
True

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 5)
kCusps = NFCusps(k); kCusps
kCusps is NFCusps(k)

Saving and loading works:

Sage

sage: loads(kCusps.dumps()) == kCusps
True

Python

>>> from sage.all import *
>>> loads(kCusps.dumps()) == kCusps
True

Sage Live

loads(kCusps.dumps()) == kCusps

class sage.modular.cusps_nf.NFCuspsSpace(number_field)[source]¶

Bases: UniqueRepresentation, Parent

The set of cusps of a number field. See NFCusps for full documentation.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 5)
sage: kCusps = NFCusps(k); kCusps
Set of all cusps of Number Field in a with defining polynomial x^2 + 5

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(5), names=('a',)); (a,) = k._first_ngens(1)
>>> kCusps = NFCusps(k); kCusps
Set of all cusps of Number Field in a with defining polynomial x^2 + 5

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 5)
kCusps = NFCusps(k); kCusps

number_field()[source]¶

Return the number field that this set of cusps is attached to.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 1)
sage: kCusps = NFCusps(k)
sage: kCusps.number_field()
Number Field in a with defining polynomial x^2 + 1

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = k._first_ngens(1)
>>> kCusps = NFCusps(k)
>>> kCusps.number_field()
Number Field in a with defining polynomial x^2 + 1

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 1)
kCusps = NFCusps(k)
kCusps.number_field()

zero()[source]¶

Return the zero cusp.

Note

This method just exists to make some general algorithms work. It is not intended that the returned cusp is an additive neutral element.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 5)
sage: kCusps = NFCusps(k)
sage: kCusps.zero()
Cusp [0: 1] of Number Field in a with defining polynomial x^2 + 5

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(5), names=('a',)); (a,) = k._first_ngens(1)
>>> kCusps = NFCusps(k)
>>> kCusps.zero()
Cusp [0: 1] of Number Field in a with defining polynomial x^2 + 5

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 5)
kCusps = NFCusps(k)
kCusps.zero()

sage.modular.cusps_nf.NFCusps_ideal_reps_for_levelN(N, nlists=1)[source]¶

Return a list of lists (nlists different lists) of prime ideals, coprime to N, representing every ideal class of the number field.

INPUT:

N – number field ideal
nlists – (default: 1) the number of lists of prime ideals we want

OUTPUT:

A list of lists of ideals representatives of the ideal classes, all coprime to N, representing every ideal.

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(5, a + 1)
sage: from sage.modular.cusps_nf import NFCusps_ideal_reps_for_levelN
sage: NFCusps_ideal_reps_for_levelN(N)
[(Fractional ideal (1), Fractional ideal (2, a + 1))]
sage: L = NFCusps_ideal_reps_for_levelN(N, 3)
sage: all(len(L[i]) == k.class_number() for i in range(len(L)))
True

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(3) + Integer(11), names=('a',)); (a,) = k._first_ngens(1)
>>> N = k.ideal(Integer(5), a + Integer(1))
>>> from sage.modular.cusps_nf import NFCusps_ideal_reps_for_levelN
>>> NFCusps_ideal_reps_for_levelN(N)
[(Fractional ideal (1), Fractional ideal (2, a + 1))]
>>> L = NFCusps_ideal_reps_for_levelN(N, Integer(3))
>>> all(len(L[i]) == k.class_number() for i in range(len(L)))
True

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^3 + 11)
N = k.ideal(5, a + 1)
from sage.modular.cusps_nf import NFCusps_ideal_reps_for_levelN
NFCusps_ideal_reps_for_levelN(N)
L = NFCusps_ideal_reps_for_levelN(N, 3)
all(len(L[i]) == k.class_number() for i in range(len(L)))

Sage

sage: k.<a> = NumberField(x^4 - x^3 - 21*x^2 + 17*x + 133)
sage: N = k.ideal(6)
sage: from sage.modular.cusps_nf import NFCusps_ideal_reps_for_levelN
sage: NFCusps_ideal_reps_for_levelN(N)
[(Fractional ideal (1),
  Fractional ideal (67, a + 17),
  Fractional ideal (127, a + 48),
  Fractional ideal (157, a - 19))]
sage: L = NFCusps_ideal_reps_for_levelN(N, 5)
sage: all(len(L[i]) == k.class_number() for i in range(len(L)))
True

Python

>>> from sage.all import *
>>> k = NumberField(x**Integer(4) - x**Integer(3) - Integer(21)*x**Integer(2) + Integer(17)*x + Integer(133), names=('a',)); (a,) = k._first_ngens(1)
>>> N = k.ideal(Integer(6))
>>> from sage.modular.cusps_nf import NFCusps_ideal_reps_for_levelN
>>> NFCusps_ideal_reps_for_levelN(N)
[(Fractional ideal (1),
  Fractional ideal (67, a + 17),
  Fractional ideal (127, a + 48),
  Fractional ideal (157, a - 19))]
>>> L = NFCusps_ideal_reps_for_levelN(N, Integer(5))
>>> all(len(L[i]) == k.class_number() for i in range(len(L)))
True

Sage Live

k.<a> = NumberField(x^4 - x^3 - 21*x^2 + 17*x + 133)
N = k.ideal(6)
from sage.modular.cusps_nf import NFCusps_ideal_reps_for_levelN
NFCusps_ideal_reps_for_levelN(N)
L = NFCusps_ideal_reps_for_levelN(N, 5)
all(len(L[i]) == k.class_number() for i in range(len(L)))

Python

>>> from sage.all import *
>>> k = NumberField(x**Integer(4) - x**Integer(3) - Integer(21)*x**Integer(2) + Integer(17)*x + Integer(133), names=('a',)); (a,) = k._first_ngens(1)
>>> N = k.ideal(Integer(6))
>>> from sage.modular.cusps_nf import NFCusps_ideal_reps_for_levelN
>>> NFCusps_ideal_reps_for_levelN(N)
[(Fractional ideal (1),
  Fractional ideal (67, a + 17),
  Fractional ideal (127, a + 48),
  Fractional ideal (157, a - 19))]
>>> L = NFCusps_ideal_reps_for_levelN(N, Integer(5))
>>> all(len(L[i]) == k.class_number() for i in range(len(L)))
True

Sage Live

k.<a> = NumberField(x^4 - x^3 - 21*x^2 + 17*x + 133)
N = k.ideal(6)
from sage.modular.cusps_nf import NFCusps_ideal_reps_for_levelN
NFCusps_ideal_reps_for_levelN(N)
L = NFCusps_ideal_reps_for_levelN(N, 5)
all(len(L[i]) == k.class_number() for i in range(len(L)))

sage.modular.cusps_nf.list_of_representatives()[source]¶

Return a list of ideals, coprime to the ideal N, representatives of the ideal classes of the corresponding number field.

Note

This list, used every time we check \(\Gamma_0(N)\) - equivalence of cusps, is cached.

INPUT:

N – an ideal of a number field

OUTPUT:

A list of ideals coprime to the ideal N, such that they are representatives of all the ideal classes of the number field.

EXAMPLES:

Sage

sage: from sage.modular.cusps_nf import list_of_representatives
sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^4 + 13*x^3 - 11)
sage: N = k.ideal(713, a + 208)
sage: L = list_of_representatives(N); L
(Fractional ideal (1),
 Fractional ideal (47, a - 9),
 Fractional ideal (53, a - 16))

Python

>>> from sage.all import *
>>> from sage.modular.cusps_nf import list_of_representatives
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(4) + Integer(13)*x**Integer(3) - Integer(11), names=('a',)); (a,) = k._first_ngens(1)
>>> N = k.ideal(Integer(713), a + Integer(208))
>>> L = list_of_representatives(N); L
(Fractional ideal (1),
 Fractional ideal (47, a - 9),
 Fractional ideal (53, a - 16))

Sage Live

from sage.modular.cusps_nf import list_of_representatives
x = polygen(ZZ, 'x')
k.<a> = NumberField(x^4 + 13*x^3 - 11)
N = k.ideal(713, a + 208)
L = list_of_representatives(N); L

sage.modular.cusps_nf.number_of_Gamma0_NFCusps(N)[source]¶

Return the total number of orbits of cusps under the action of the congruence subgroup \(\Gamma_0(N)\).

INPUT:

N – a number field ideal

OUTPUT: integer; the number of orbits of cusps under Gamma0(N)-action

EXAMPLES:

Sage

sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(2, a+1)
sage: from sage.modular.cusps_nf import number_of_Gamma0_NFCusps
sage: number_of_Gamma0_NFCusps(N)
4
sage: L = Gamma0_NFCusps(N)
sage: len(L) == number_of_Gamma0_NFCusps(N)
True
sage: k.<a> = NumberField(x^2 + 7)
sage: N = k.ideal(9)
sage: number_of_Gamma0_NFCusps(N)
6
sage: N = k.ideal(a*9 + 7)
sage: number_of_Gamma0_NFCusps(N)
24

Python

>>> from sage.all import *
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(3) + Integer(11), names=('a',)); (a,) = k._first_ngens(1)
>>> N = k.ideal(Integer(2), a+Integer(1))
>>> from sage.modular.cusps_nf import number_of_Gamma0_NFCusps
>>> number_of_Gamma0_NFCusps(N)
4
>>> L = Gamma0_NFCusps(N)
>>> len(L) == number_of_Gamma0_NFCusps(N)
True
>>> k = NumberField(x**Integer(2) + Integer(7), names=('a',)); (a,) = k._first_ngens(1)
>>> N = k.ideal(Integer(9))
>>> number_of_Gamma0_NFCusps(N)
6
>>> N = k.ideal(a*Integer(9) + Integer(7))
>>> number_of_Gamma0_NFCusps(N)
24

Sage Live

x = polygen(ZZ, 'x')
k.<a> = NumberField(x^3 + 11)
N = k.ideal(2, a+1)
from sage.modular.cusps_nf import number_of_Gamma0_NFCusps
number_of_Gamma0_NFCusps(N)
L = Gamma0_NFCusps(N)
len(L) == number_of_Gamma0_NFCusps(N)
k.<a> = NumberField(x^2 + 7)
N = k.ideal(9)
number_of_Gamma0_NFCusps(N)
N = k.ideal(a*9 + 7)
number_of_Gamma0_NFCusps(N)

sage.modular.cusps_nf.units_mod_ideal(I)[source]¶

Return integral elements of the number field representing the images of the global units modulo the ideal I.

INPUT:

I – number field ideal

OUTPUT:

A list of integral elements of the number field representing the images of the global units modulo the ideal I. Elements of the list might be equivalent to each other mod I.

EXAMPLES:

Sage

sage: from sage.modular.cusps_nf import units_mod_ideal
sage: x = polygen(ZZ, 'x')
sage: k.<a> = NumberField(x^2 + 1)
sage: I = k.ideal(a + 1)
sage: units_mod_ideal(I)
[1]
sage: I = k.ideal(3)
sage: units_mod_ideal(I)
[1, a, -1, -a]

Python

>>> from sage.all import *
>>> from sage.modular.cusps_nf import units_mod_ideal
>>> x = polygen(ZZ, 'x')
>>> k = NumberField(x**Integer(2) + Integer(1), names=('a',)); (a,) = k._first_ngens(1)
>>> I = k.ideal(a + Integer(1))
>>> units_mod_ideal(I)
[1]
>>> I = k.ideal(Integer(3))
>>> units_mod_ideal(I)
[1, a, -1, -a]

Sage Live

from sage.modular.cusps_nf import units_mod_ideal
x = polygen(ZZ, 'x')
k.<a> = NumberField(x^2 + 1)
I = k.ideal(a + 1)
units_mod_ideal(I)
I = k.ideal(3)
units_mod_ideal(I)

Sage

sage: from sage.modular.cusps_nf import units_mod_ideal
sage: k.<a> = NumberField(x^3 + 11)
sage: k.unit_group()
Unit group with structure C2 x Z of
 Number Field in a with defining polynomial x^3 + 11
sage: I = k.ideal(5, a + 1)
sage: units_mod_ideal(I)
[1,
-2*a^2 - 4*a + 1,
...]

Python

>>> from sage.all import *
>>> from sage.modular.cusps_nf import units_mod_ideal
>>> k = NumberField(x**Integer(3) + Integer(11), names=('a',)); (a,) = k._first_ngens(1)
>>> k.unit_group()
Unit group with structure C2 x Z of
 Number Field in a with defining polynomial x^3 + 11
>>> I = k.ideal(Integer(5), a + Integer(1))
>>> units_mod_ideal(I)
[1,
-2*a^2 - 4*a + 1,
...]

Sage Live

from sage.modular.cusps_nf import units_mod_ideal
k.<a> = NumberField(x^3 + 11)
k.unit_group()
I = k.ideal(5, a + 1)
units_mod_ideal(I)

Python

>>> from sage.all import *
>>> from sage.modular.cusps_nf import units_mod_ideal
>>> k = NumberField(x**Integer(3) + Integer(11), names=('a',)); (a,) = k._first_ngens(1)
>>> k.unit_group()
Unit group with structure C2 x Z of
 Number Field in a with defining polynomial x^3 + 11
>>> I = k.ideal(Integer(5), a + Integer(1))
>>> units_mod_ideal(I)
[1,
-2*a^2 - 4*a + 1,
...]

Sage Live

from sage.modular.cusps_nf import units_mod_ideal
k.<a> = NumberField(x^3 + 11)
k.unit_group()
I = k.ideal(5, a + 1)
units_mod_ideal(I)

Sage

sage: from sage.modular.cusps_nf import units_mod_ideal
sage: k.<a> = NumberField(x^4 - x^3 -21*x^2 + 17*x + 133)
sage: k.unit_group()
Unit group with structure C6 x Z of
 Number Field in a with defining polynomial x^4 - x^3 - 21*x^2 + 17*x + 133
sage: I = k.ideal(3)
sage: U = units_mod_ideal(I)
sage: all(U[j].is_unit() and (U[j] not in I) for j in range(len(U)))
True

Python

>>> from sage.all import *
>>> from sage.modular.cusps_nf import units_mod_ideal
>>> k = NumberField(x**Integer(4) - x**Integer(3) -Integer(21)*x**Integer(2) + Integer(17)*x + Integer(133), names=('a',)); (a,) = k._first_ngens(1)
>>> k.unit_group()
Unit group with structure C6 x Z of
 Number Field in a with defining polynomial x^4 - x^3 - 21*x^2 + 17*x + 133
>>> I = k.ideal(Integer(3))
>>> U = units_mod_ideal(I)
>>> all(U[j].is_unit() and (U[j] not in I) for j in range(len(U)))
True

Sage Live

from sage.modular.cusps_nf import units_mod_ideal
k.<a> = NumberField(x^4 - x^3 -21*x^2 + 17*x + 133)
k.unit_group()
I = k.ideal(3)
U = units_mod_ideal(I)
all(U[j].is_unit() and (U[j] not in I) for j in range(len(U)))

Python

>>> from sage.all import *
>>> from sage.modular.cusps_nf import units_mod_ideal
>>> k = NumberField(x**Integer(4) - x**Integer(3) -Integer(21)*x**Integer(2) + Integer(17)*x + Integer(133), names=('a',)); (a,) = k._first_ngens(1)
>>> k.unit_group()
Unit group with structure C6 x Z of
 Number Field in a with defining polynomial x^4 - x^3 - 21*x^2 + 17*x + 133
>>> I = k.ideal(Integer(3))
>>> U = units_mod_ideal(I)
>>> all(U[j].is_unit() and (U[j] not in I) for j in range(len(U)))
True

Sage Live

from sage.modular.cusps_nf import units_mod_ideal
k.<a> = NumberField(x^4 - x^3 -21*x^2 + 17*x + 133)
k.unit_group()
I = k.ideal(3)
U = units_mod_ideal(I)
all(U[j].is_unit() and (U[j] not in I) for j in range(len(U)))