Elliptic curves over the rational numbers with semi-abelian reduction and two-division points

We classify elliptic curves over the rationals whose Neron 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. (c) 2020 Elsevier Inc. All rights reserved.