A multiple zeta value over Fq[t] (abbreviated as MZV) is Eulerian if it is a rational multiple of a power of the Carlitz period. By the work of Anderson and Thakur, every MZV is closely related to the periods of an explicitly constructed t-motive, whose motivic Galois group, coming from Papanikolas’ Tannakian duality theory, is a linear algebraic group over Fq(t). A conjecture of Lara and Thakur on Eulerian MZV’s implies that their corresponding motivic Galois groups are of dimension 1. Assuming Lara-Thakur’s conjecture, in this talk, we give the explicit defining equations of these 1 dimensional motivic Galois groups.