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 href="https://doi.org/10.1016/j.jalgebra.2006.01.015">"A computational solution to a question by Beauville on the invariants of the binary quintic"</a>, J. Algebra **303** (2006) 771-788. The preprint version is <a href="https://arxiv.org/abs/math/0508421">here</a>.