  Many rationally parametrized elliptic modular equations are derived. Each
comes from a family of elliptic curves attached to a genus-zero congruence
subgroup $\Gamma_0(N)$, as an algebraic transformation of elliptic curve
periods, parametrized by a Hauptmodul (function field generator). The periods
satisfy a Picard-Fuchs equation, of hypergeometric, Heun, or more general type;
so the new modular equations are algebraic transformations of special
functions. When N=4,3,2 they are modular transformations of Ramanujan's
elliptic integrals of signatures 2,3,4. This gives a modern interpretation to
his theories of integrals to alternative bases: they are attached to certain
families of elliptic curves. His anomalous theory of signature 6 turns out to
fit into a general Gauss-Manin rather than a Picard-Fuchs framework.
