8.6.2.1 ManinSymbol Objects

class ManinSymbol
A Manin symbol $ [X^i\cdot Y^{k-2-i},(u,v)]$ .
ManinSymbol( parent, t)

Create a Manin symbol $ [X^i*Y^{k-2-i},(u,v)]$ , where $ k$ is the weight.

INPUT:
    parent -- ManinSymbolList
    t -- a 3-tuple (i,u,v) of int's.

Instances of class ManinSymbol have the following functions (in addition to inherited functions and special functions):

apply,$  $ copy,$  $ endpoints,$  $ i,$  $ lift_to_sl2z,$  $ modular_symbol_rep,$  $ parent,$  $ tuple,$  $ u,$  $ v,$  $ weight

Further documentation:

apply( a, b, c, d)

Return the image of self under the matrix [a,b;c,d].

INPUT:
    a, b, c, d -- integers
    
OUTPUT:
    a list of pairs (alpha_i, x_i), where each alpha_i is an
integer, 
    x_i is a Manin symbol, and the sum alpha_i*x_i is the image
of
    self under the right action of the matrix [a,b;c,d].
    Here the right action of g=[a,b;c,d] on a Manin symbol
[P(X,Y),(u,v)]
    is [P(aX+bY,cX+dY),(u,v)*g].

endpoints( [N=None])

Returns cusps alpha, beta such that this Manin symbol, viewed as a symbol for level N, is $ X^i*Y^{k-2-i} \{alpha, beta\}$ .

lift_to_sl2z( N)

If this Manin symbol is (c,d) viewed modulo N, 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).

modular_symbol_rep( )

Returns a representation of self as a formal sum of modular symbols. (The result is not cached.)

Instances of class ManinSymbol also have the following special functions:

__cmp__,$  $ __mul__

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