Lists of Manin symbols (elements of \(\mathbb{P}^1(R/N)\)) over number fields¶
Lists of elements of \(\mathbb{P}^1(R/N)\) where \(R\) is the ring of integers of a number field \(K\) and \(N\) is an integral ideal.
AUTHORS:
Maite Aranes (2009): Initial version
EXAMPLES:
We define a P1NFList:
sage: k.<a> = NumberField(x^3 + 11)
sage: N = k.ideal(5, a^2 - a + 1)
sage: P = P1NFList(N); P
The projective line over the ring of integers modulo the Fractional ideal (5, a^2 - a + 1)
List operations with the P1NFList:
sage: len(P)
26
sage: [p for p in P]
[M-symbol (0: 1) of level Fractional ideal (5, a^2 - a + 1),
...
M-symbol (1: 2*a^2 + 2*a) of level Fractional ideal (5, a^2 - a + 1)]
The elements of the P1NFList are M-symbols:
sage: type(P[2])
<class 'sage.modular.modsym.p1list_nf.MSymbol'>
Definition of MSymbols:
sage: alpha = MSymbol(N, 3, a^2); alpha
M-symbol (3: a^2) of level Fractional ideal (5, a^2 - a + 1)
Find the index of the class of an M-Symbol \((c: d)\) in the list:
sage: i = P.index(alpha)
sage: P[i].c*alpha.d - P[i].d*alpha.c in N
True
Lift an MSymbol to a matrix in \(SL(2, R)\):
sage: alpha = MSymbol(N, a + 2, 3*a^2)
sage: alpha.lift_to_sl2_Ok()
[-3*a^2 + a + 12, 25*a^2 - 50*a + 100, a + 2, a^2 - 3*a + 3]
sage: Ok = k.ring_of_integers()
sage: M = Matrix(Ok, 2, alpha.lift_to_sl2_Ok())
sage: det(M)
1
sage: M[1][1] - alpha.d in N
True
Lift an MSymbol from P1NFList to a matrix in \(SL(2, R)\)
sage: P[3]
M-symbol (1: -2*a) of level Fractional ideal (5, a^2 - a + 1)
sage: P.lift_to_sl2_Ok(3)
[0, -1, 1, -2*a]
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class
sage.modular.modsym.p1list_nf.
MSymbol
(N, c, d=None, check=True)¶ Bases:
sage.structure.sage_object.SageObject
The constructor for an M-symbol over a number field.
INPUT:
N
– integral ideal (the modulus or level).c
– integral element of the underlying number field or an MSymbol of level N.d
– (optional) when present, it must be an integral element such that <c> + <d> + N = R, where R is the corresponding ring of integers.check
– bool (default True). Ifcheck=False
the constructor does not check the condition <c> + <d> + N = R.
OUTPUT:
An M-symbol modulo the given ideal N, i.e. an element of the projective line \(\mathbb{P}^1(R/N)\), where R is the ring of integers of the underlying number field.
EXAMPLES:
sage: k.<a> = NumberField(x^3 + 11) sage: N = k.ideal(a + 1, 2) sage: MSymbol(N, 3, a^2 + 1) M-symbol (3: a^2 + 1) of level Fractional ideal (2, a + 1)
We can give a tuple as input:
sage: MSymbol(N, (1, 0)) M-symbol (1: 0) of level Fractional ideal (2, a + 1)
We get an error if <c>, <d> and N are not coprime:
sage: MSymbol(N, 2*a, a - 1) Traceback (most recent call last): ... ValueError: (2*a, a - 1) is not an element of P1(R/N). sage: MSymbol(N, (0, 0)) Traceback (most recent call last): ... ValueError: (0, 0) is not an element of P1(R/N).
Saving and loading works:
sage: alpha = MSymbol(N, 3, a^2 + 1) sage: loads(dumps(alpha))==alpha True
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N
()¶ Return the level or modulus of this MSymbol.
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 23) sage: N = k.ideal(3, a - 1) sage: alpha = MSymbol(N, 3, a) sage: alpha.N() Fractional ideal (3, 1/2*a - 1/2)
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c
¶ Return the first coefficient of the M-symbol.
EXAMPLES:
sage: k.<a> = NumberField(x^3 + 11) sage: N = k.ideal(a + 1, 2) sage: alpha = MSymbol(N, 3, a^2 + 1) sage: alpha.c # indirect doctest 3
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d
¶ Return the second coefficient of the M-symbol.
EXAMPLES:
sage: k.<a> = NumberField(x^3 + 11) sage: N = k.ideal(a + 1, 2) sage: alpha = MSymbol(N, 3, a^2 + 1) sage: alpha.d # indirect doctest a^2 + 1
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lift_to_sl2_Ok
()¶ Lift the MSymbol to an element of \(SL(2, Ok)\), where \(Ok\) is the ring of integers of the corresponding number field.
OUTPUT:
A list of integral elements \([a, b, c', d']\) that are the entries of a 2x2 matrix with determinant 1. The lower two entries are congruent (modulo the level) to the coefficients \(c, d\) of the MSymbol self.
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 23) sage: N = k.ideal(3, a - 1) sage: alpha = MSymbol(N, 3*a + 1, a) sage: alpha.lift_to_sl2_Ok() [0, -1, 1, a]
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normalize
(with_scalar=False)¶ Return a normalized MSymbol (a canonical representative of an element of \(\mathbb{P}^1(R/N)\) ) equivalent to
self
.INPUT:
with_scalar
– bool (default False)
OUTPUT:
(only if
with_scalar=True
) a transforming scalar \(u\), such that \((u*c', u*d')\) is congruent to \((c: d)\) (mod \(N\)), where \((c: d)\) are the coefficients ofself
and \(N\) is the level.a normalized MSymbol (c’: d’) equivalent to
self
.
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 23) sage: N = k.ideal(3, a - 1) sage: alpha1 = MSymbol(N, 3, a); alpha1 M-symbol (3: a) of level Fractional ideal (3, 1/2*a - 1/2) sage: alpha1.normalize() M-symbol (0: 1) of level Fractional ideal (3, 1/2*a - 1/2) sage: alpha2 = MSymbol(N, 4, a + 1) sage: alpha2.normalize() M-symbol (1: -a) of level Fractional ideal (3, 1/2*a - 1/2)
We get the scaling factor by setting
with_scalar=True
:sage: alpha1.normalize(with_scalar=True) (a, M-symbol (0: 1) of level Fractional ideal (3, 1/2*a - 1/2)) sage: r, beta1 = alpha1.normalize(with_scalar=True) sage: r*beta1.c - alpha1.c in N True sage: r*beta1.d - alpha1.d in N True sage: r, beta2 = alpha2.normalize(with_scalar=True) sage: r*beta2.c - alpha2.c in N True sage: r*beta2.d - alpha2.d in N True
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tuple
()¶ Return the MSymbol as a list (c, d).
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 23) sage: N = k.ideal(3, a - 1) sage: alpha = MSymbol(N, 3, a); alpha M-symbol (3: a) of level Fractional ideal (3, 1/2*a - 1/2) sage: alpha.tuple() (3, a)
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class
sage.modular.modsym.p1list_nf.
P1NFList
(N)¶ Bases:
sage.structure.sage_object.SageObject
The class for \(\mathbb{P}^1(R/N)\), the projective line modulo \(N\), where \(R\) is the ring of integers of a number field \(K\) and \(N\) is an integral ideal.
INPUT:
N
- integral ideal (the modulus or level).
OUTPUT:
A P1NFList object representing \(\mathbb{P}^1(R/N)\).
EXAMPLES:
sage: k.<a> = NumberField(x^3 + 11) sage: N = k.ideal(5, a + 1) sage: P = P1NFList(N); P The projective line over the ring of integers modulo the Fractional ideal (5, a + 1)
Saving and loading works.
sage: loads(dumps(P)) == P True
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N
()¶ Return the level or modulus of this P1NFList.
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 31) sage: N = k.ideal(5, a + 3) sage: P = P1NFList(N) sage: P.N() Fractional ideal (5, 1/2*a + 3/2)
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apply_J_epsilon
(i, e1, e2=1)¶ Apply the matrix \(J_{\epsilon}\) = [e1, 0, 0, e2] to the i-th M-Symbol of the list.
e1, e2 are units of the underlying number field.
INPUT:
i
– integere1
– unite2
– unit (default 1)
OUTPUT:
integer – the index of the M-Symbol obtained by the right action of the matrix \(J_{\epsilon}\) = [e1, 0, 0, e2] on the i-th M-Symbol.
EXAMPLES:
sage: k.<a> = NumberField(x^3 + 11) sage: N = k.ideal(5, a + 1) sage: P = P1NFList(N) sage: u = k.unit_group().gens_values(); u [-1, 2*a^2 + 4*a - 1] sage: P.apply_J_epsilon(4, -1) 2 sage: P.apply_J_epsilon(4, u[0], u[1]) 1
sage: k.<a> = NumberField(x^4 + 13*x - 7) sage: N = k.ideal(a + 1) sage: P = P1NFList(N) sage: u = k.unit_group().gens_values(); u [-1, a^3 + a^2 + a + 12, a^3 + 3*a^2 - 1] sage: P.apply_J_epsilon(3, u[2]^2)==P.apply_J_epsilon(P.apply_J_epsilon(3, u[2]),u[2]) True
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apply_S
(i)¶ Applies the matrix S = [0, -1, 1, 0] to the i-th M-Symbol of the list.
INPUT:
i
– integer
OUTPUT:
integer – the index of the M-Symbol obtained by the right action of the matrix S = [0, -1, 1, 0] on the i-th M-Symbol.
EXAMPLES:
sage: k.<a> = NumberField(x^3 + 11) sage: N = k.ideal(5, a + 1) sage: P = P1NFList(N) sage: j = P.apply_S(P.index_of_normalized_pair(1, 0)) sage: P[j] M-symbol (0: 1) of level Fractional ideal (5, a + 1)
We test that S has order 2:
sage: j = randint(0,len(P)-1) sage: P.apply_S(P.apply_S(j))==j True
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apply_TS
(i)¶ Applies the matrix TS = [1, -1, 0, 1] to the i-th M-Symbol of the list.
INPUT:
i
– integer
OUTPUT:
integer – the index of the M-Symbol obtained by the right action of the matrix TS = [1, -1, 0, 1] on the i-th M-Symbol.
EXAMPLES:
sage: k.<a> = NumberField(x^3 + 11) sage: N = k.ideal(5, a + 1) sage: P = P1NFList(N) sage: P.apply_TS(3) 2
We test that TS has order 3:
sage: j = randint(0,len(P)-1) sage: P.apply_TS(P.apply_TS(P.apply_TS(j)))==j True
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apply_T_alpha
(i, alpha=1)¶ Applies the matrix T_alpha = [1, alpha, 0, 1] to the i-th M-Symbol of the list.
INPUT:
i
– integeralpha
– element of the corresponding ring of integers(default 1)
OUTPUT:
integer – the index of the M-Symbol obtained by the right action of the matrix T_alpha = [1, alpha, 0, 1] on the i-th M-Symbol.
EXAMPLES:
sage: k.<a> = NumberField(x^3 + 11) sage: N = k.ideal(5, a + 1) sage: P = P1NFList(N) sage: P.apply_T_alpha(4, a^ 2 - 2) 3
We test that T_a*T_b = T_(a+b):
sage: P.apply_T_alpha(3, a^2 - 2)==P.apply_T_alpha(P.apply_T_alpha(3,a^2),-2) True
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index
(c, d=None, with_scalar=False)¶ Return the index of the class of the pair \((c, d)\) in the fixed list of representatives of \(\mathbb{P}^1(R/N)\).
INPUT:
c
– integral element of the corresponding number field, or an MSymbol.d
– (optional) when present, it must be an integral element of the number field such that \((c, d)\) defines an M-symbol of level \(N\).with_scalar
– bool (default False)
OUTPUT:
u
- the normalizing scalar (only ifwith_scalar=True
)i
- the index of \((c, d)\) in the list.
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 31) sage: N = k.ideal(5, a + 3) sage: P = P1NFList(N) sage: P.index(3,a) 5 sage: P[5]==MSymbol(N, 3, a).normalize() True
We can give an MSymbol as input:
sage: alpha = MSymbol(N, 3, a) sage: P.index(alpha) 5
We cannot look for the class of an MSymbol of a different level:
sage: M = k.ideal(a + 1) sage: beta = MSymbol(M, 0, 1) sage: P.index(beta) Traceback (most recent call last): ... ValueError: The MSymbol is of a different level
If we are interested in the transforming scalar:
sage: alpha = MSymbol(N, 3, a) sage: P.index(alpha, with_scalar=True) (-a, 5) sage: u, i = P.index(alpha, with_scalar=True) sage: (u*P[i].c - alpha.c in N) and (u*P[i].d - alpha.d in N) True
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index_of_normalized_pair
(c, d=None)¶ Return the index of the class \((c, d)\) in the fixed list of representatives of \(\mathbb(P)^1(R/N)\).
INPUT:
c
– integral element of the corresponding number field, or a normalized MSymbol.d
– (optional) when present, it must be an integral element of the number field such that \((c, d)\) defines a normalized M-symbol of level \(N\).
OUTPUT:
i
- the index of \((c, d)\) in the list.
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 31) sage: N = k.ideal(5, a + 3) sage: P = P1NFList(N) sage: P.index_of_normalized_pair(1, 0) 3 sage: j = randint(0,len(P)-1) sage: P.index_of_normalized_pair(P[j])==j True
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lift_to_sl2_Ok
(i)¶ Lift the \(i\)-th element of this P1NFList to an element of \(SL(2, R)\), where \(R\) is the ring of integers of the corresponding number field.
INPUT:
i
- integer (index of the element to lift)
OUTPUT:
If the \(i\)-th element is \((c : d)\), the function returns a list of integral elements \([a, b, c', d']\) that defines a 2x2 matrix with determinant 1 and such that \(c=c'\) (mod \(N\)) and \(d=d'\) (mod \(N\)).
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 23) sage: N = k.ideal(3) sage: P = P1NFList(N) sage: len(P) 16 sage: P[5] M-symbol (1/2*a + 1/2: -a) of level Fractional ideal (3) sage: P.lift_to_sl2_Ok(5) [-a, 2*a - 2, 1/2*a + 1/2, -a]
sage: Ok = k.ring_of_integers() sage: L = [Matrix(Ok, 2, P.lift_to_sl2_Ok(i)) for i in range(len(P))] sage: all(det(L[i]) == 1 for i in range(len(L))) True
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list
()¶ Return the underlying list of this P1NFList object.
EXAMPLES:
sage: k.<a> = NumberField(x^3 + 11) sage: N = k.ideal(5, a+1) sage: P = P1NFList(N) sage: type(P) <class 'sage.modular.modsym.p1list_nf.P1NFList'> sage: type(P.list()) <... 'list'>
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normalize
(c, d=None, with_scalar=False)¶ Return a normalised element of \(\mathbb{P}^1(R/N)\).
INPUT:
c
– integral element of the underlying number field, or an MSymbol.d
– (optional) when present, it must be an integral element of the number field such that \((c, d)\) defines an M-symbol of level \(N\).with_scalar
– bool (default False)
OUTPUT:
(only if
with_scalar=True
) a transforming scalar \(u\), such that \((u*c', u*d')\) is congruent to \((c: d)\) (mod \(N\)).a normalized MSymbol (c’: d’) equivalent to \((c: d)\).
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 31) sage: N = k.ideal(5, a + 3) sage: P = P1NFList(N) sage: P.normalize(3, a) M-symbol (1: 2*a) of level Fractional ideal (5, 1/2*a + 3/2)
We can use an MSymbol as input:
sage: alpha = MSymbol(N, 3, a) sage: P.normalize(alpha) M-symbol (1: 2*a) of level Fractional ideal (5, 1/2*a + 3/2)
If we are interested in the normalizing scalar:
sage: P.normalize(alpha, with_scalar=True) (-a, M-symbol (1: 2*a) of level Fractional ideal (5, 1/2*a + 3/2)) sage: r, beta = P.normalize(alpha, with_scalar=True) sage: (r*beta.c - alpha.c in N) and (r*beta.d - alpha.d in N) True
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sage.modular.modsym.p1list_nf.
P1NFList_clear_level_cache
()¶ Clear the global cache of data for the level ideals.
EXAMPLES:
sage: k.<a> = NumberField(x^3 + 11) sage: N = k.ideal(a+1) sage: alpha = MSymbol(N, 2*a^2, 5) sage: alpha.normalize() M-symbol (-4*a^2: 5*a^2) of level Fractional ideal (a + 1) sage: sage.modular.modsym.p1list_nf._level_cache {Fractional ideal (a + 1): (...)} sage: sage.modular.modsym.p1list_nf.P1NFList_clear_level_cache() sage: sage.modular.modsym.p1list_nf._level_cache {}
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sage.modular.modsym.p1list_nf.
lift_to_sl2_Ok
(N, c, d)¶ Lift a pair (c, d) to an element of \(SL(2, O_k)\), where \(O_k\) is the ring of integers of the corresponding number field.
INPUT:
N
– number field idealc
– integral element of the number fieldd
– integral element of the number field
OUTPUT:
A list [a, b, c’, d’] of integral elements that are the entries of a 2x2 matrix with determinant 1. The lower two entries are congruent to c, d modulo the ideal \(N\).
EXAMPLES:
sage: from sage.modular.modsym.p1list_nf import lift_to_sl2_Ok sage: k.<a> = NumberField(x^2 + 23) sage: Ok = k.ring_of_integers() sage: N = k.ideal(3) sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 1, a)) sage: det(M) 1 sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 0, a)) sage: det(M) 1 sage: (M[1][0] in N) and (M[1][1] - a in N) True sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 0, 0)) Traceback (most recent call last): ... ValueError: Cannot lift (0, 0) to an element of Sl2(Ok).
sage: k.<a> = NumberField(x^3 + 11) sage: Ok = k.ring_of_integers() sage: N = k.ideal(3, a - 1) sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 2*a, 0)) sage: det(M) 1 sage: (M[1][0] - 2*a in N) and (M[1][1] in N) True sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 4*a^2, a + 1)) sage: det(M) 1 sage: (M[1][0] - 4*a^2 in N) and (M[1][1] - (a+1) in N) True
sage: k.<a> = NumberField(x^4 - x^3 -21*x^2 + 17*x + 133) sage: Ok = k.ring_of_integers() sage: N = k.ideal(7, a) sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 0, a^2 - 1)) sage: det(M) 1 sage: (M[1][0] in N) and (M[1][1] - (a^2-1) in N) True sage: M = Matrix(Ok, 2, lift_to_sl2_Ok(N, 0, 7)) Traceback (most recent call last): ... ValueError: <0> + <7> and the Fractional ideal (7, a) are not coprime.
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sage.modular.modsym.p1list_nf.
make_coprime
(N, c, d)¶ Return (c, d’) so d’ is congruent to d modulo N, and such that c and d’ are coprime (<c> + <d’> = R).
INPUT:
N
– number field idealc
– integral element of the number fieldd
– integral element of the number field
OUTPUT:
A pair (c, d’) where c, d’ are integral elements of the corresponding number field, with d’ congruent to d mod N, and such that <c> + <d’> = R (R being the corresponding ring of integers).
EXAMPLES:
sage: from sage.modular.modsym.p1list_nf import make_coprime sage: k.<a> = NumberField(x^2 + 23) sage: N = k.ideal(3, a - 1) sage: c = 2*a; d = a + 1 sage: N.is_coprime(k.ideal(c, d)) True sage: k.ideal(c).is_coprime(d) False sage: c, dp = make_coprime(N, c, d) sage: k.ideal(c).is_coprime(dp) True
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sage.modular.modsym.p1list_nf.
p1NFlist
(N)¶ Return a list of the normalized elements of \(\mathbb{P}^1(R/N)\), where \(N\) is an integral ideal.
INPUT:
N
- integral ideal (the level or modulus).
EXAMPLES:
sage: k.<a> = NumberField(x^2 + 23) sage: N = k.ideal(3) sage: from sage.modular.modsym.p1list_nf import p1NFlist, psi sage: len(p1NFlist(N))==psi(N) True
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sage.modular.modsym.p1list_nf.
psi
(N)¶ The index \([\Gamma : \Gamma_0(N)]\), where \(\Gamma = GL(2, R)\) for \(R\) the corresponding ring of integers, and \(\Gamma_0(N)\) standard congruence subgroup.
EXAMPLES:
sage: from sage.modular.modsym.p1list_nf import psi sage: k.<a> = NumberField(x^2 + 23) sage: N = k.ideal(3, a - 1) sage: psi(N) 4
sage: k.<a> = NumberField(x^2 + 23) sage: N = k.ideal(5) sage: psi(N) 26