TITLE: Is there a classification of pointed nodal genus 1 curves?
QUESTION [6 upvotes]: 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 $y^2 = x^3 + x^2$)?

REPLY [3 votes]: I am just posting my comment as an answer.  The closed substack $\Delta$ of $\overline{\mathcal{M}}_{1,1}$ parameterizing pointed, nodal, stable curves of arithmetic genus $1$ is naturally equivalent to the classifying stack $BC_2$, where $C_2$ is the cyclic group of order $2$.  Thus, for every scheme $S$ (or algebraic space), the set of equivalence classes of $1$-morphisms from $S$ to $\Delta$ is bijective to the set of isomorphism classes of $C_2$-torsors over $S$, i.e., finite, degree $2$, étale covers of $S$ (possibly disconnected).  For a field $k$ of characteristic $\neq 2$, the set of isomorphism classes is bijective to $k^\times/(k^\times)^2$.