The $j$ invariant is

$j=\frac{S^3}{S^3-27T^2}$

where

$S=a-\frac{bd}{4}+\frac{c^2}{12}$

and

$T=\frac{ac}{6}+\frac{bcd}{48}-\frac{c^3}{216}-\frac{ad^2}{16}-\frac{b^2}{16}$

for more details see my article J. Algebra 303 (2006) 771-788.