Any pointed nodal (ie, proper semistable with a specified rational point lying in the smooth locus) curve of arithmetic genus 1 over a field $k$ must be irreducible and has precisely 1 node, which must be rational over the base field, and its normalization must be $\mathbb{P}^1_k$.

Over an algebraically closed field, all such curves are isomorphic to the compactification of the plane curve $y^2 = x^3 + x^2$, say with the marked point at $P = (0,0)$.

Over a general field $k$, is there a classification of pointed nodal curves of arithmetic genus 1 (such curves would essentially be twists of $y^2 = x^3 + x^2$)?

`nodal' and`

semistable', do you require that every P^1 has at least 2 special points? The terminology is not uniformly applied; I think`semistable' is very ambiguous (cf Liu/de Jong vs the enumerative geometry literature), and`

nodal' usually puts no restriction on the automorphism group, in which case your curve need not be irreducible. $\endgroup$