TITLE: explicit formula for the j-invariant of binary quartic form
QUESTION [6 upvotes]: A binary quartic form
$aX^4+bX^3Y+cX^2Y^2+dXY^3+Y^4$
decomposes as a product of linear factors $Y-t_jX$, $j=1,...,4$. 
I would like to have an explicit formula for symmetrization of the crossratio of $t_j$.

REPLY [11 votes]: 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.