from eis_series import eisenstein_series_qexp
from vm_basis import delta_qexp
from sage.rings.all import QQ
def j_invariant_qexp(prec=10, K=QQ):
r"""
Return the $q$-expansion of the $j$-invariant to
precision prec in the field $K$.
WARNING: Stupid algorithm -- we divide by Delta, which is slow.
EXAMPLES:
sage: j_invariant_qexp(4)
q^-1 + 744 + 196884*q + 21493760*q^2 + 864299970*q^3 + O(q^4)
sage: j_invariant_qexp(2)
q^-1 + 744 + 196884*q + O(q^2)
sage: j_invariant_qexp(100, GF(2))
q^-1 + q^7 + q^15 + q^31 + q^47 + q^55 + q^71 + q^87 + O(q^100)
"""
if prec <= -1:
raise ValueError, "the prec must be nonnegative."
prec += 2
g6 = -504*eisenstein_series_qexp(6, prec, K=QQ)
Delta = delta_qexp(prec).change_ring(QQ)
j = (g6*g6) * (~Delta) + 1728
if K != QQ:
return j.change_ring(K)
else:
return j
