Third kind elliptic integrals and 1-motives

In [4] we have showed that the Generalized Grothendieck's Period Conjecture applied to 1-motives, whose underlying semi-abelian variety is a product of elliptic curves and of tori, is equivalent to a transcendental conjecture involving elliptic integrals of the first and second kind, and logarithms of complex numbers. In this paper we investigate the Generalized Grothendieck's Period Conjecture in the case of a 1-motive M whose underlying semi-abelian variety is a non trivial extension of a product of elliptic curves by a torus. This will imply the introduction of elliptic integrals of the third kind for the computation of the periods of M and therefore the Generalized Grothendieck's Period Conjecture applied to M will be equivalent to a transcendental conjecture involving elliptic integrals of the first, second and third kind.