On supercurves of genus 1 with an underlying odd spin structure

We study the standard family of supercurves of genus 1 with an underlying odd spin structures. We give a simple algebraic description of this family and of the compactified family of stable supercurves with one Neveu-Schwarz puncture. We also describe the Gauss-Manin connection on the first de Rham cohomology of this family and compute the superperiods of global differentials.