Dimension of the motivic Galois group of a 1-motive

We compute the dimension of the motivic Galois group of a 1-motive M defined over the field of complex numbers, expressing it explicitly in terms of the rank of the multiplicative group generated by the points defining M. As an application, we obtain a new formulation of the Grothendieck–André periods Conjecture in the setting of 1-motives.