Prime Numbers

7.5 Counting points on elliptic curves 363
Obtain a root j ∈ Fp of S, via Algorithm 2.3.10;
c = j(j − 1728) −1 mod p;
r = −3c mod p;
s =2c mod p;
7. [Return two curve-parameter pairs]
return {(a, b)} = {(r, s), (rg 2 mod p, sg 3 mod p)};
What the Atkin–Morain method prescribes is that for D = −4, −3 the
governing cubics are given in terms of a quadratic nonresidue g, whichis
also a cubic nonresidue in the case D = −3, by
y 2 = x 3 − g k x, k =0, 1, 2, 3,
y 2 = x 3 − g k , k =0, 1, 2, 3, 4, 5,
respectively (i.e., there are respectively 4, 6 isomorphism classes of curves for
these two D values); while for other discriminants D the relevant curve and
its twist are
y 2 = x 3 − 3cg 2k x +2cg 3k , k =0, 1,
where c is given as in Step [Continuation for D