27.9 List of Elements of $ P^1(\mathbf{Z}/N\mathbf{Z})$

Module: sage.modular.modsym.p1list

Module-level Functions

_make_p1list( n)

p1_normalize( N, u, v)

p1_normalize_int( N, u, v)

p1_normalize_int(N, u, v):

Computes the canonical representative of $ \mathbf{P}^1(\mathbf{Z}/N\mathbf{Z})$ equivalent to $ (u,v)$ along with a transforming scalar.

INPUT:
    N -- an integer
    u -- an integer
    v -- an integer
    
OUTPUT:
    If gcd(u,v,N) = 1, then returns
         uu -- an integer
         vv -- an integer
         ss -- an integer
         such that (ss*uu, ss*vv) is equivalent to (u,v) mod N
    and if gcd(u,v,N) != 1, returns
         0, 0, 0

p1_normalize_llong( N, u, v)

p1_normalize_llong(N, u, v):

Computes the canonical representative of $ \mathbf{P}^1(\mathbf{Z}/N\mathbf{Z})$ equivalent to $ (u,v)$ along with a transforming scalar.

INPUT:
    N -- an integer
    u -- an integer
    v -- an integer
OUTPUT:
    If gcd(u,v,N) = 1, then returns
         uu -- an integer
         vv -- an integer
         ss -- an integer
         such that (ss*uu, ss*vv) is equivalent to (u,v) mod N
    and if gcd(u,v,N) != 1, returns
         0, 0, 0

p1list( int N)

p1list_int( int N)

p1list_int(int N):

Make a list of the normalized elements of $ \mathbf{P}^1(\mathbf{Z}/N\mathbf{Z})$ .

p1list_llong( int N)

p1list_llong(int N):

Make a list of the normalized elements of $ \mathbf{P}^1(\mathbf{Z}/N\mathbf{Z})$ .

search( )

Return (True,i) where i is such that v[i] == x if there is such an i, or (False,j) otherwise, where j is the position that a should be inserted so that v remains sorted.

INPUT:
    v -- a list, which is assumed sorted
    x -- Python object
OUTPUT:
    bool, int

Class: export

class export

Class: P1List

class P1List

Functions: apply_I,$  $ apply_S,$  $ apply_T,$  $ index,$  $ index_and_scalar,$  $ index_of_normalized_pair,$  $ list,$  $ N,$  $ normalize,$  $ normalize_with_scalar

index( self, int u, int v)

Returns the index of the class of $ (u,v)$ in the fixed list of representatives of $ \mathbf{P}^1(\mathbf{Z}/N\mathbf{Z})$ .

INPUT:
    u, v -- integers, with GCD(u,v,N)=1.

OUTPUT:
    i -- the index of u, v, in the P^1 list.

index_and_scalar( self, int u, int v)

Returns the index of the class of $ (u,v)$ in the fixed list of representatives of $ PP^1(Z/NZ)$ .

INPUT:
    u, v -- integers, with GCD(u,v,N)=1.
    
OUTPUT:
    i -- the index of u, v, in the P^1 list.
    s -- scalar that transforms normalized form to u,v

index_of_normalized_pair( self, int u, int v)

Returns the index of the class of $ (u,v)$ in the fixed list of representatives of $ \mathbf{P}^1(\mathbf{Z}/N\mathbf{Z})$ .

INPUT:
    u, v -- integers, with GCD(u,v,N)=1 normalized so they lie in the list.
OUTPUT:
    i -- the index of u, v, in the P^1 list.

normalize_with_scalar( self, int u, int v)

normalize_with_scalar(self, int u, int v)

INPUT:
    u, v -- integers, with GCD(u,v,N)=1.
    
OUTPUT:
    uu, vv -- integers of *normalized* rep
    ss -- scalar such that (ss*uu, ss*vv) = (u,v) mod N

Special Functions: __getitem__,$  $ __getslice__,$  $ __len__,$  $ __reduce__,$  $ __repr__

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