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Principally polarized abelian surfaces with prescribed real multiplication(RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus2 curves correspond to rational points on the Hilbert modular surfaces viatheir Jacobians, but the converse is not true. We give a simple genericdescription of which rational moduli points correspond to rational curves, aswell as give associated Weierstrass models, in the case of RM by the ring ofintegers of $\mathbb{Q}(\sqrt{5})$. To prove this, we provide some techniquesfor reducing quadratic forms over polynomial rings.

Moduli for rational genus 2 curves with real multiplication for discriminant 5