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 $t\_j$.



explicit formula for the j-invariant of binary quartic form
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"](https://www.sciencedirect.com/science/article/pii/S0021869306000287), J. Algebra **303** (2006) 771-788. The preprint version is [here](https://arxiv.org/abs/math/0508421).


