# When is the period of elliptic curve over the rationals transcendental?

Given an elliptic curve $E/\mathbf{Q}$, when is its period transcendental/algebraic?

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This is a subject of an old (from 1970s) theorem by Chudnovsky. You have to distinguish the CM and non-CM cases. BTW, this is "transcendental" rather than "algebraic" number theory. –  Wadim Zudilin Jun 16 '10 at 13:29

On p. 304 of "Contributions to the theory of transcendental numbers" by Gregory Chudnovsky (avalaible from google books) one finds a consequence of Theorem 1.26 which states (even more than) that if $E$ has a complex multiplication in a number field, then any period is transcendental.