Let $E$ be an elliptic curve over $\mathbb{Q}$ with Weierstrass equation: $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$ with coefficients $a_i\in\mathbb{Q}$ Let: \begin{eqnarray} \nonumber b_2 &=& a_1^2+4a_2,\\ \nonumber b_4 &=& 2a_4+a_1a_3,\\ \nonumber b_6 &=& a_3^2+4a_6,\\ \nonumber b_8 &=& a_1^2a_6+4a_2a_6-a_1a_3a_4+a_3^2a_2-a_4^2,\\ \nonumber c_4 &=& b_2^2-24b_4,\\ \nonumber c_6 &=& -b_2^3+36b_2b_4-216b_6 \end{eqnarray}

Example:

If $E$ has a Weierstrass equation in the simplified form $y^2=x^3+Ax+B$ then $$ \Delta=-16(4A^3+27B^2),\quad j=-\frac{1728(4A)^3}{\Delta}$$

Note: The discriminant $\Delta$ coincides in this case with the usual notion of discriminant of the polynomial $x^3+Ax+B$