Abstract
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.
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