j-invariant of an elliptic curve (reviewed)

The $j$-invariant of an elliptic curve $E$ defined over a field $K$ is an invariant of the isomorphism class of $E$ over $\overline{K}$. If the Weierstrass equation of $E$ is \[ y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6, \] then its $j$-invariant is given by \[ j = \frac{c_4^3}{\Delta} \] where \[\begin{aligned} b_2 &= a_1^2 + 4 a_2\\ b_4 &= 2a_4 + a_1 a_3\\ b_6 &= a_3^2 + 4 a_6 \\ b_8 &= a_1^2 a_6 + 4 a_2 a_6 - a_1 a_3 a_4 + a_2 a_3^2 - a_4^2\\ c_4 &= b_2^2 - 24b_4 \end{aligned}\] and \[ \Delta=-b_2^2b_8 - 8 b_4^3 -27 b_6 ^2 + 9 b_2 b_4 b_6 \]

is the discriminant of $E.$