Cummins and Pauli have calculated generators for the function fields of all congruence subgroups of $\text{PSL}_2(\mathbb{F}_p)$ \text{PSL}_2(\mathbb{Z})$of genus$\le 24$in: http://www.mathstat.concordia.ca/faculty/cummins/congruence/ I haven't looked at this for a few months but I believe that the companion paper http://www.emis.de/journals/EM/expmath/volumes/12/12.2/pp243_255.pdf discusses the generators. In the meantime there is a paper by Yifan Yang "Defining equations of modular curves" Advances in Mathematics Volume 204, Issue 2, 20 August 2006, Pages 481-508 which gives tables of equations for many modular curves, and discusses a methodology for finding "good" equations (i.e. those with small coefficients and a small number of terms in the defining polynomials) 2 added 552 characters in body Cummins and Pauli have calculated generators for the function fields of all congruence subgroups of$\text{PSL}_2(\mathbb{F}_p)$of genus$\le 24$in: http://www.mathstat.concordia.ca/faculty/cummins/congruence/ I haven't looked at this for a few months but I believe that the companion paper http://www.emis.de/journals/EM/expmath/volumes/12/12.2/pp243_255.pdf discusses the generators. In the meantime there is a paper by Yifan Yang "Defining equations of modular curves" Advances in Mathematics Volume 204, Issue 2, 20 August 2006, Pages 481-508 which gives tables of equations for many modular curves, and discusses a methodology for finding "good" equations (i.e. those with small coefficients and a small number of terms in the defining polynomials) 1 Cummins and Pauli have calculated generators for the function fields of all congruence subgroups of$\text{PSL}_2(\mathbb{F}_p)$of genus$\le 24\$ in: