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§4. Endomorphisms and Complex Multiplication 243

Finally we consider the determination of all elliptic curves isogenous to a given
elliptic curve E with complex multiplication.
(4.9) Remarks. For an imaginary quadratic field K, let R denote the ring of integers
in K. For a subring A of R note that rk(A) = rk(R) = 2 if and only if as abelian
groups A is of finite index in R. Every such subring A has the form R f = Z + fR,
where f is an integer f ≥ 1 called the conductor of the order R f = A.
We denote by Pic(A) the projective class group. It consists of isomorphism
classes of projective modules L over A of rank 1 with abelian group structure
given by tensor product of projective modules of rank 1. There is an embedding
L → Q ⊗ Z L = K ⊗ A L = V ,and V is a one-dimensional vector space over K.
A choice of basis element gives an embedding L → K as a fractional ideal, and two
such embeddings differ up to multiplication by an element of K. Hence a second
interpretation of Pic(A) is as fractional ideal classes of A in its field of fractions K
when A = R f .
For a number field K and an order A contained in the ring of integes R of K,the
group Pic(A) is finite and its cardinality is the class number of A or of K in the case
A = R. For the A = R f = Z + fR we denote by h K, f = #Pic(R f ), the cardinality
of Pic.
(4.10) Theorem. Let K ⊂ C be a quadratic imaginary field. For each class [L] in
Pic(R f ), choose a representative Lof [L] and an embedding L → K ⊂ C.The
function that assigns to [L] the elliptic curve E L with E L (C) = C/Lis a bijection
of Pic(R f ) onto the set of isomorphism classes of elliptic curves E over C with
End(E) = R f .
Proof. Let E be an elliptic curve over C with E(C) = C/L where L can be chosen to
have a nonzero element in common with K. Since R f L = L, where R f = End(E),
we have L ⊂ K as a functional idea, and this ideal is defined up to scalar multiple.
(L) = End(C/L) = R f .
This proves the theorem since End R f
The ring R f = Z + fR is called the order with conductor f in an imaginary
quadratic field K.

(4.11) Proposition. The j-invariant of C/Lfor any [L] in Pic(R f ), denoted j(L) =
f (C/L), is an algebraic number of degree ≤ h K, f .

Proof. The group Aut(C) acts on the finite set J K, f of all j(L) for [L] ∈ Pic(R f ),
and this means that these numbers are algebraic numbers of degree less than the
cardinality of the finite set.
(4.12) Remark. The question of j values of complex multiplication curves will be
considered further in Chapter 13. For example j(L) is an algebraic integer of degree
exactly h f,K and Aut(C) acts transitively on the set J K, f . For further information
see Serre, “Complex Multiplication,” in the Cassels and Frohlich book [1967]. We
include a table of the j values of L = R f for the cases where h K, f = 1 from Serre
[1967]. Classically these values appear in Weber, Algebra III,see §§125–128 and
Tabelle VI.