Cyclic codes constitute a significant subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, by analyzing the solutions of certain equations over \(\mathbb {F}_{5^m}\) and using the multivariate method, we propose eight classes of optimal quinary cyclic codes \(\mathcal {C}_{(1, e, \frac{5^m-1}{2})}\) and prove that our new optimal quinary cyclic codes are inequivalent to the known ones. Moreover, we show that the quinary cyclic codes \(\mathcal {C}_{(\frac{5^m+1}{2}, \frac{5^m-1}{2}+e, 5^m-1)}\) and \(\mathcal {C}_{(1, e, \frac{5^m-1}{2})}\) have the same optimality. As a result, we can get new optimal quinary cyclic codes from known ones. In addition, we reveal the quinary cyclic codes \(\mathcal {C}_{(\frac{5^m+1}{2}, \frac{5^m-1}{2}+e, \frac{5^m-1}{2})}\) and \(\mathcal {C}_{(1,e,\frac{5^m-1}{2})}\) have the same parameters for \(e\equiv 3\pmod 4\) .
