On the automorphism group of quotient modular curves

In this article, we determine the automorphism group of all the quotient modular curves of the modular curve X 0 ( p q ) , where p , q are two distinct primes. In obtaining such results, we provide different insights to compute the automorphism group for any quotient modular curve, which are very effective when the level of the curve is square-free. In particular, in the case where the level of the quotient curve is non square-free, we would mention that we present an unfamiliar automorphism of order 3 for the genus 4 curve X 0 ⁎ ( 25 ⋅ 11 ) defined over Q [ 5 ] .