Spacelike mean curvature 1 surfaces of genus 1 with two ends in de Sitter 3-space

We give a mathematical foundation for, and numerical demonstration of, the existence of mean curvature 1 surfaces of genus 1 with either two elliptic ends or two hyperbolic ends in de Sitter 3-space. An end of a mean curvature 1 surface is an 'elliptic end' (respectively a 'hyperbolic end') if the monodromy matrix at the end is diagonalizable with eigenvalues in the unit circle (respectively in the reals). Although the existence of the surfaces is numerical, the types of ends are mathematically determined.