  We classify elliptic curves over the rationals whose N\'eron 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.
