## Degree of isogenies of elliptic curves with complex multiplication.(English)Zbl 0943.11031

Let $$E$$ be an elliptic curve with complex multiplication such that $$\operatorname {End} E\cong O_f$$, where $$O_f= \mathbb{Z}+ fO_K$$ and $$K= \mathbb{Q} (\sqrt{D})$$. Let $$F:= \mathbb{Q}(j(E))$$ and suppose that $$E$$ is defined over $$F$$. The author gives a complete description of those elliptic curves $$E'$$ that are $$F$$-isogenous to $$E$$ and calculates all possible degrees $$N$$ of cyclic isogenies $$E'\to E$$ defined over $$F$$. The degree $$N$$ of a cyclic isogeny $$E\to E'$$ is (roughly speaking) the number of divisors of $$f^2 d_K$$ where $$d_K$$ denotes the discriminant of $$K$$.

### MSC:

 11G05 Elliptic curves over global fields 11G15 Complex multiplication and moduli of abelian varieties

### Keywords:

elliptic curve; complex multiplication; degree; cyclic isogeny