Rational and Elliptic Parametrizations ofQ-Curves Original Research Article

We describe explicit parametrizations of the rational points ofX*(N), the algebraic curve obtained as quotient of the modular curveX0(N) by the groupB(N) generated by the Atkin–Lehner involutions, wheneverNis square-free and the curve is rational or elliptic. By taking into account the moduli interpretation ofX*(N), along with a standard “boundedness” conjecture, we obtain all theQ-isogeny classes ofQ-curves except for a finite set.