@cached_function
def E_polynomial(g, n=None):
    r"""
    Returns the known values of E_g.
    
    If n is specified, return a symmetricp polynomial in
    variables x0, ..., x(n-1).

    EXAMPLES::

        sage: E_polynomial(0)
        -e[]
        sage: E_polynomial(1)
        1/12*e[] + 1/12*e[2]
        sage: E_polynomial(1,1)
        1/12
        sage: E_polynomial(1,2)
        1/12*x0*x1 + 1/12
    """
    e = SymmetricFunctions(QQ).elementary()
    if g == 0:
        result = - e.one()
    if g == 1:
        result = 1/12*(e.one() + e[2])
    if g == 2:
        result = -1/240*e.one() + 1/192*e[1, 1] - 1/3840*e[1, 1, 1, 1] - 1/144*e[2] + 1/1440*e[2, 1, 1]\
               - 1/360*e[2, 2] + 1/288*e[3, 1] - 1/720*e[4]
    if g == 3:
        result = 1/1008*e.one() - 1/768*e[1, 1] + 19/73728*e[1, 1, 1, 1] - 19/4128768*e[1, 1, 1, 1, 1, 1]\
        + 1/480*e[2] - 19/23040*e[2, 1, 1] + 19/860160*e[2, 1, 1, 1, 1] + 1/720*e[2, 2] - 1/7680*e[2, 2, 1, 1]\
        + 1/3360*e[2, 2, 2] - 7/5760*e[3, 1] + 17/107520*e[3, 1, 1, 1] - 23/40320*e[3, 2, 1] - 1/10080*e[3, 3]\
        + 1/1440*e[4] + 11/80640*e[4, 1, 1] + 1/2016*e[4, 2] - 1/2240*e[5, 1] + 1/5040*e[6]
    if g == 4:
        # R.<a_0,a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9,a_10> = PolynomialRing(QQ)
        R.<r1, r2> = PolynomialRing(QQ)
        e = SymmetricFunctions(R).elementary()
        result = -1/1440*e.one() + 5/5184*e[1, 1] - 7/25920*e[1, 1, 1, 1] + 751/40310784*e[1, 1, 1, 1, 1, 1] - 407/2418647040*e[1, 1, 1, 1, 1, 1, 1, 1] - 5/3024*e[2] + 1/1080*e[2, 1, 1] - 751/7838208*e[2, 1, 1, 1, 1] + 407/352719360*e[2, 1, 1, 1, 1, 1, 1] - 1/720*e[2, 2] + 73/217728*e[2, 2, 1, 1] - 89/9797760*e[2, 2, 1, 1, 1, 1] - 1/2016*e[2, 2, 2] + 11/272160*e[2, 2, 2, 1, 1] - 1/15120*e[2, 2, 2, 2] + 1/960*e[3, 1] - 391/1306368*e[3, 1, 1, 1] + 839/78382080*e[3, 1, 1, 1, 1, 1] + 29/36288*e[3, 2, 1] - 103/1306368*e[3, 2, 1, 1, 1] + 31/181440*e[3, 2, 2, 1] + 1/6048*e[3, 3] - 23/362880*e[3, 3, 1, 1] + 1/15120*e[3, 3, 2] - 1/1440*e[4] + (-5/217728)*e[4, 1, 1] + (67/2612736)*e[4, 1, 1, 1, 1] - 5/6048*e[4, 2] + 1/217728*e[4, 2, 1, 1] - 1/4320*e[4, 2, 2] + 1/30240*e[4, 3, 1] - 1/15120*e[4, 4] + (r2-313/181440)*e[5, 1] + (-1/36*r2)*e[5, 1, 1, 1] + 53/181440*e[5, 2, 1] + 1/10080*e[5, 3] - 1/3024*e[6] - 29/362880*e[6, 1, 1] - 1/4320*e[6, 2] + 13/60480*e[7, 1] - 1/10080*e[8]
    if g > 4:
        raise ValueError(f'E_{g} not calculated yet')
    
    if n is None:
        return result
    return result.expand(n)