We classify elliptic curves over the rationals whose Néron model over the integers is semi-abelian, with good reduction at $p=2$, and whose Mordell–Weil group contains an element of order two that stays non-trivial at $p=2$. Furthermore, we describe those curves where the element of order two is narrow, or where another element of order two exists, and also express our findings in terms of Deligne–Mumford stacks of pointed curves of genus one.

中文翻译：

---

带半阿贝尔约简和二分点的有理数上的椭圆曲线

我们对有理数上的椭圆曲线进行分类，其整数上的 Néron 模型是半阿贝尔的，在 $磷=2$，并且其 Mordell-Weil 群包含一个二阶元素，该元素在 $磷=2$. 此外，我们描述了那些二阶元素很窄的曲线，或者存在另一个二阶元素的曲线，并且还用 Deligne-Mumford 堆栈的一属尖曲线来表达我们的发现。