Mumford-Tate groups of 1-motives and Weil pairing

We show how the geometry of a 1-motive $M$ (that is existence of endomorphisms and relations between the points defining it) determines the dimension of its motivic Galois group $\mathcal{G}{\mathrm{a}\mathrm{l}}_{\mathrm{m}\mathrm{o}\mathrm{t}}(M)$. Fixing periods matrices ${\mathrm{\Pi }}_M$ and ${\mathrm{\Pi }}_{M^{\ast }}$ associated respectively to a 1-motive $M$ and to its Cartier dual $M^{\ast },$ we describe the action of the Mumford-Tate group of $M$ on these matrices. In the semi-elliptic case, according to the geometry of $M$ we classify polynomial relations between the periods of $M$ and we compute exhaustively the matrices representing the Mumford-Tate group of $M$. This representation brings new light on Grothendieck periods conjecture in the case of 1-motives.