Generic models for genus 2 curves with real multiplication

Explicit models of families of genus 2 curves with multiplication by $\sqrt{D}$ are known for $D=2,3,5$. We obtain generic models for genus 2 curves over $\mathbb{Q}$ with real multiplication in 12 new cases, including all fundamental discriminants $D<40$. A key step in our proof is to develop an algorithm for minimisation of conic bundles fibred over ${\mathbb{P}}^2$. We apply this algorithm to simplify the equations for the Mestre conic associated to the generic point on the Hilbert modular surface of fundamental discriminant $D<100$ computed by Elkies–Kumar.