Take the tour ×  A binary quartic form   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 $t_j$.
share|improve this question Do you mean the j-invariant of the
elliptic curve $y^2=ax^4+bx^3+cx^2+dx+1$? –  Robin Chapman Jul 7
'10 at 13:54 Yes, that is exactly what I am looking for. – 
David Marín Jul 7 '10 at 14:00 If you have access to a computer
algebra system, you can do the following. Let $\xi$ denote a
solution to $f(1,\xi)=0$ where $f$ is your quartic. Then
$f(X,Y+\xi X)=b'X^3Y+\cdots+Y^4$. The elliptic curve is now
isomorphic to $y^2=b'x^3+c'x^2+d'x+1$. Transform it to the usual
Weierstrass form and take the $j$-invariant. Note that $b'$ etc.
will have $\xi$s in them, but they should all cancel out via the
equation $f(1,\xi)=0$ in the final result. –  Robin Chapman Jul
7 '10 at 14:23 add comment  1 Answer  up vote 7 down vote
accepted  The $j$ invariant is       for more details see my
article J. Algebra 30