Fix a quadratic imaginary field . Let be the set of -isomorphism classes of elliptic curves with . By the above results we may also view as the set of lattices with .

If is a fractional ideal,
then
is a lattice in .
For the elliptic curve
we have

because is an -module by definition. Since rescaling a lattice produces an isomorphic elliptic curve, for any nonzero the fractional ideals and define the same elements of .

The *class group* is the group
of fractional ideals modulo principal fractional ideals.
If is a fractional ideal,
denote by
its ideal class in the class
group of .
We have a natural map

which sends to .

- is a lattice in ,
- We have .
- We have if and only if .

William 2007-05-25