Module: sage.modular.modsym.p1list
List of Elements of
Module-level Functions
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Return a list of Python ints [a,b,c',d'] that are the entries of a 2x2 matrix with determinant 1 and lower two entries congruent to c,d modulo N.
sage: lift_to_sl2z(2,3,6) [1, 1, 2, 3] sage: lift_to_sl2z(15,6,24) [-2, -17, 15, 126] sage: lift_to_sl2z(15,6,2400000) [-2L, -320001L, 15L, 2400006L]
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Lift a pair (c, d) to an element of SL(2, Z)
(c,d) is assumed to be an element of P1(Z/NZ). This function computes and returns a list [a, b, c', d'] that defines a 2x2 matrix, with determinant 1 and integer entries, such that c=c'(mod N) and d=d'(mod N).
EXAMPLES
sage: lift_to_sl2z_int(2,6,11) [1, 8, 2, 17] sage: m=Matrix(Integers(),2,2,lift_to_sl2z_int(2,6,11)) sage: m [ 1 8] [ 2 17]
Author: Justin Walker
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Lift a pair (c, d) to an element of SL(2, Z)
(c,d) is assumed to be an element of P1(Z/NZ). This function computes and returns a list [a, b, c', d'] that defines a 2x2 matrix, with determinant 1 and integer entries, such that c=c'(mod N) and d=d'(mod N).
EXAMPLES
sage: lift_to_sl2z_llong(2,6,11) [1L, 8L, 2L, 17L] sage: m=Matrix(Integers(),2,2,lift_to_sl2z_llong(2,6,11)) sage: m [ 1 8] [ 2 17]
Author: Justin Walker
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p1_normalize_int(N, u, v):
Computes the canonical representative of
equivalent
to
along with a transforming scalar.
Input:
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p1_normalize_llong(N, u, v):
Computes the canonical representative of
equivalent
to
along with a transforming scalar.
Input:
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p1list_int(int N):
Make a list of the normalized elements of
.
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p1list_llong(int N):
Make a list of the normalized elements of
.
Class: export
Class: P1List
sage: P = P1List(12); P The projective line over the integers modulo 12 sage: list(P) [(0, 1), (1, 0), (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (1, 9), (1, 10), (1, 11), (2, 1), (2, 3), (2, 5), (3, 1), (3, 2), (3, 4), (3, 7), (4, 1), (4, 3), (4, 5), (6, 1)]
Saving and loading works.
sage: loads(dumps(P)) == P True
Functions: apply_I,
apply_S,
apply_T,
index,
index_and_scalar,
index_of_normalized_pair,
lift_to_sl2z,
list,
N,
normalize,
normalize_with_scalar
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Returns the index of the class of
in the fixed list of
representatives of
.
Input:
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Returns the index of the class of
in the fixed list of
representatives of
.
Input:
) |
Returns the index of the class of
in the fixed list of
representatives of
.
Input:
) |
Lift an element of P1 to an element of SL(2,Z)
If the ith P1 element is (c,d), this function computes and returns a list [a,b, c',d'] that defines a 2x2 matrix with determinant 1 and integer entries, such that c=c'(mod N) and d=d'(mod N).
EXAMPLES
sage: p=P1List(11)
sage: p.list()[3] (1, 2)
sage: p.lift_to_sl2z(3) [0, -1, 1, 2]
Author: Justin Walker
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normalize_with_scalar(self, int u, int v)
Input:
Special Functions: __cmp__,
__getitem__,
__getslice__,
__init__,
__len__,
__reduce__,
__repr__
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