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 "A computational solution to a question by Beauville on the invariants of the binary quintic", J. Algebra 303 (2006) 771-788. The preprint version is here.
