The j-invariant of the binary quartic form $aX^4+bX^3Y+cX^2Y^2+dXY^3+eY^4$ can be expressed by the two basic invariants I and J of the binary quartic form.
\begin{align*}
j&=\frac{I^3}{\Delta}=\frac{I^3}{I^3-27J^2}\\
I&=\begin{vmatrix}
 a & \frac{c}{6} \\
 \frac{c}{6} & e \\
\end{vmatrix} -4 \begin{vmatrix}
 \frac{b}{4} & \frac{c}{6} \\
 \frac{c}{6} & \frac{d}{4} \\
\end{vmatrix}=a e-\frac{b d}{4}+\frac{c^2}{12}\\
J&=\begin{vmatrix}
 a & \frac{b}{4} & \frac{c}{6} \\
 \frac{b}{4} & \frac{c}{6} & \frac{d}{4} \\
 \frac{c}{6} & \frac{d}{4} & e \\
\end{vmatrix}=\frac{a c e}{6}-\frac{a d^2}{16}-\frac{b^2 e}{16}+\frac{b c d}{48}-\frac{c^3}{216}\\
\end{align*}

An Introduction to Invariants and Moduli 

(Cambridge Studies in Advanced Mathematics, Series Number 81)(Page 27)

https://www.math.ens.psl.eu/~benoist/refs/Mukai.pdf

Invariants of binary and ternary forms (Page 18)

https://ritzenth.pages.math.cnrs.fr/web//TEDI/tedi5-ritzenthaler.pdf