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The modular equation of order b is a polynomial relation between j(z) and j(z/b), which has astronomically large coefficients even for small values of b. Fricke showed that a two-valued relation exists for 37 small values of b. This relation would have much smaller coefficients and would also be convenient for finding singular moduli. Although Fricke produced no two-valued relations explicitly (no doubt because of the tedious amount of algebraic manipulation), they are now found by use ofdoi:10.1090/s0025-5718-1988-0935079-4 fatcat:dyd6egrfjfadjeb2zrxtowspxy