Assuming that I know the size of the base field ($q$), size of the prime order subgroup ($r$), and embedding degree of the curve ($k$) of a pairing friendly elliptic curve, I would like to instantiate a curve in both Weierstrass and Edwards form.

How do I find A, B (also c, d) when I know the q, r, and k?

$y^2 = x^3 + Ax + B$

$u^2 + v^2 = c^2 (1 + d \cdot u^2 v^2)$

Assuming that I know the size of the base field ($q$), size of the prime order subgroup ($r$), and embedding degree of the curve ($k$) of a pairing friendly elliptic curve ($E$), I would like to instantiate a curve in both Weierstrass and Edwards form.

How do I find $A, B$ (also $c, d$) when I know the $q, r,$ and $k$?

$E: y^2 = x^3 + Ax + B$

$E: u^2 + v^2 = c^2 (1 + d \cdot u^2 v^2)$

where $r$ divides $\#E(\mathcal{F}_q)$ and $k$ is the smallest integer such that $r$ divides $q^k - 1$.