In this short paper, we propose a variant of the Number Field Sieve to compute discrete logarithms in medium characteristic finite fields. We propose an algorithm that combines two recent ideas, namely the Multiple variant of the Number Field Sieve taking advantage of a large number of number fields in the sieving phase and the Conjugation Method giving a new polynomial selection for the classical Number Field Sieve. The asymptotic complexity of our improved algorithm is L_Q (1/3, (8(9 + 4√6)/15)^(1/3)), where F_Q is the target finite field and (8(9+4√6)/15)^(1/3)) ≈ 2.156. This has to be compared with the complexity of the previous state-of-the-art algorithm for medium characteristic finite fields, the Number Field Sieve with Conjugation Method, that has a complexity of approximately L_Q (1/3, 2.201).