  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
${\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}(M)$. Fixing periods matrices $\Pi_M$
and $\Pi_{M^*}$ associated respectively to a 1-motive $M$ and to its Cartier
dual $M^*,$ 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.
