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

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. 


You need to assume E defined over the algebraic numbers, or else "the" period makes no sense; and then there are two basic periods, except in the complex multiplication case where the ratio will be algebraic. The first transcendence results were due to Schneider and Baker. These were developed by Coates and Masser. Probably anything you need is in Masser's thesis, which includes results on the quasiperiods too. There are many further results, but I think no real surprises. (Review by Moreno at http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183540631 of Masser's Elliptic Functions and Transcendence.) 

