The integration of differential forms furnishes an isomorphism between the De Rham and the Hodge realizations of a $1$-motive $M$. The coefficients of the matrix representing this isomorphism are the so-called "periods" of $M$.  In the semi-elliptic case (i.e. the underlying extension of the $1$-motive is an extension of an elliptic curve by the multiplicative group), we compute explicitly these periods.  

If the $1$-motive $M$ is defined over an algebraically closed field, Grothendieck's conjecture asserts that the transcendence degree of the field generated by the periods is equal to the dimension of the motivic Galois group of $M$. If we denote by $I$ the ideal generated by the polynomial relations between the periods, we have that "the numbers of periods of $M$ minus the rank of the ideal $I$ is equal to the dimension of the motivic Galois group of $M$", that is a decrease in the dimension of the motivic Galois group is equivalent to an increase of the rank of the ideal $I$. We list the geometrical phenomena which imply the decrease in the dimension of the motivic Galois group and in each case we compute the polynomials which generate the corresponding ideal $I$.