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We show how the geometry of a 1-motive $M$ (that is existence ofendomorphisms and relations between the points defining it) determines thedimension 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 Cartierdual $M^*,$ we describe the action of the Mumford-Tate group of $M$ on thesematrices. In the semi-elliptic case, according to the geometry of $M$ weclassify polynomial relations between the periods of $M$ and we computeexhaustively the matrices representing the Mumford-Tate group of $M$. Thisrepresentation brings new light on Grothendieck periods conjecture in the caseof 1-motives.

Mumford-Tate groups of 1-motives and Weil pairing