I am reading this paper (Section 4) here they want to construct an elliptic curve over $\mathbb{F}_q$ whose order has discriminant $'x'$.
They do that in two steps :
- Firstly they calculate the j-invariant of such specific curve.
- Then from j-invariant they find the equation of elliptic curve (in Generalized Wiestrass form)
They say if $a,b,c \in \mathbb{Z}$ satisfy $x=b^2-4ac$ then $j(\frac{b+i\sqrt{|x|}}{2a})$ will be the required j-invariant.
I tried to consider this elliptic curve as a lattice generated by $\{ 1,\tau\}$ and tried to find $\tau$ using the fact that Endomorphism ring will send Lattice points to lattice points but I am unable to conclude the result specified in the paper.