How to test if a given elliptic curve has complex multiplication

Question

Is there a general, reasonably easy to understand, algorithm for testing whether an elliptic curve has CM? For example, consider the curve $y^2=x^3+\frac{27}{1727}x+\frac{54}{1727}$

This has j-invariant 1, which in particular is an algebraic integer. Is there a good way of seeing that this doesn't have CM?

[I would prefer an answer which gives a general procedure rather than a trick which works for that specific curve.]

Answer 1

I am not sure of all the details, but there must exist such an algorithm as Sage knows how to check for complex multiplication.

sage: E = EllipticCurve([27/1727, 54/1727])

sage: E.has_rational_cm()
False

And you can get all the $j$-invariants of elliptic curves defined over $\mathbb{Q}$ with complex multiplication this way

sage: cm_j_invariants(QQ)

[-262537412640768000, -147197952000, -884736000, -12288000, -884736, -32768, -3375,
 0, 1728, 8000, 54000, 287496, 16581375]

which atleast would give you very simple test that would tell you that your specific $j$-invariant-one-curve do not have complex multiplication.

Maybe with this information you can look into how the algorithm really works.