A binary quartic form
$aX^4+bX^3Y+cX^2Y^2+dXY^3+Y^4$
decomposes as a product of linear factors $Yt_jX$, $j=1,...,4$. I would like to have an explicit formula for symmetrization of the crossratio of $t_j$.
A binary quartic form $aX^4+bX^3Y+cX^2Y^2+dXY^3+Y^4$ decomposes as a product of linear factors $Yt_jX$, $j=1,...,4$. I would like to have an explicit formula for symmetrization of the crossratio of $t_j$. 


The $j$ invariant is $j=\frac{S^3}{S^327T^2}$ 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) 771788. 

