The problem of classifying elliptic curves over $\mathbb Q$ with a given discriminant has received much attention. The analogous problem for genus $2$ curves has only been tackled when the absolute discriminant is a power of $2$. In this article, we classify genus $2$ curves C defined over ${\mathbb Q}$ with at least two rational Weierstrass points and whose absolute discriminant is an odd prime. In fact, we show that such a curve C must be isomorphic to a specialization of one of finitely many $1$-parameter families of genus $2$ curves. In particular, we provide genus $2$ analogues to Neumann–Setzer families of elliptic curves over the rationals.

使用给定判别式对$\mathbb Q$上的椭圆曲线进行分类的问题受到了广泛关注。仅当绝对判别式是$2$的幂时，才解决了genus $2$曲线的类似问题。在本文中，我们对在${\mathbb Q}$上定义的genus $2$曲线C进行分类，其中至少有两个有理 Weierstrass 点，并且其绝对判别式是奇素数。事实上，我们证明这样的曲线C必须同构于属$2$曲线的有限多个$1$参数族之一的特化。特别是，我们为有理数上的 Neumann-Setzer 椭圆曲线族提供了$2$ 的类似物。