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 Proposition 1.25., Page 427 Theorem 11.44.)
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