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# Elements of Arithmetic Subgroups

Elements of Arithmetic Subgroups

class `sage.modular.arithgroup.arithgroup_element.``ArithmeticSubgroupElement`

  Bases: `sage.structure.element.MultiplicativeGroupElement`

  An element of the group \({\rm SL}_2(\ZZ)\), i.e. a 2x2 integer matrix of determinant 1.

  `a`()

    Return the upper left entry of self.

    EXAMPLES:

        sage: Gamma0(13)([7,1,13,2]).a()
        7

  `acton`(z)

    Return the result of the action of self on z as a fractional linear transformation.

    EXAMPLES:

        sage: G = Gamma0(15)
        sage: g = G([1, 2, 15, 31])

    An example of g acting on a symbolic variable:

        sage: z = var('z')
        sage: g.acton(z)
        (z + 2)/(15*z + 31)

    An example involving the Gaussian numbers:

        sage: K.<i> = NumberField(x^2 + 1)
        sage: g.acton(i)
        1/1186*i + 77/1186

    An example with complex numbers:

        sage: C.<i> = ComplexField()
        sage: g.acton(i)
        0.0649241146711636 + 0.000843170320404721*I

    An example with the cusp infinity:

        sage: g.acton(infinity)
        1/15

    An example which maps a finite cusp to infinity:

        sage: g.acton(-31/15)
        +Infinity

    Note that when acting on instances of cusps the return value is still a rational number or infinity (Note the presence of ‘+’, which does not show up for cusp instances):

        sage: g.acton(Cusp(-31/15))
        +Infinity

    TESTS:

    We cover the remaining case, i.e., infinity mapped to infinity:

        sage: G([1, 4, 0, 1]).acton(infinity)
        +Infinity

  `b`()

    Return the upper right entry of self.

    EXAMPLES:

        sage: Gamma0(13)([7,1,13,2]).b()
        1

  `c`()

    Return the lower left entry of self.

    EXAMPLES:

        sage: Gamma0(13)([7,1,13,2]).c()
        13

  `d`()

    Return the lower right entry of self.

    EXAMPLES:

        sage: Gamma0(13)([7,1,13,2]).d()
        2

  `det`()

    Return the determinant of self, which is always 1.

    EXAMPLES:

        sage: Gamma1(11)([12,11,-11,-10]).det()
        1

  `determinant`()

    Return the determinant of self, which is always 1.

    EXAMPLES:

        sage: Gamma0(691)([1,0,691,1]).determinant()
        1

  `matrix`()

    Return the matrix corresponding to self.

    EXAMPLES:

        sage: x = Gamma1(3)([4,5,3,4]) ; x
        [4 5]
        [3 4]
        sage: x.matrix()
        [4 5]
        [3 4]
        sage: type(x.matrix())
        <type 'sage.matrix.matrix_integer_2x2.Matrix_integer_2x2'>

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