Given a real number, how do I produce an elliptic curve with j-invariant equal to that number?

Question

I have formula for computing the j-invariant but I was wondering if given number $j$, is there a formula for getting a curve $y^2=x^3+a_2x^2+a_4x+a_6$ with j-invariant j?

Answer 1

I found it!

\begin{eqnarray*} Y^2 + XY = X^3 -\dfrac{36}{j-1728}X-\dfrac{1}{j-1728} \end{eqnarray*}