  Surfaces that evolve by mean curvature flow develop singularities. These
singularities can be modeled by self-shrinkers, surfaces that shrink by
dilations under the flow. Singularities modeled on classical self-shrinkers,
namely spheres and cylinders, are stable under perturbations of the flow. In
contrast, singularities modeled on other self-shrinkers, such as the Angenent
torus, are unstable: perturbing the flow will generally change the kind of
singularity. One can measure the degree of instability by computing the Morse
index of the self-shrinker, viewed as a critical point of an appropriate
functional.
  In this paper, we present a numerical method for computing the index of
rotationally symmetric self-shrinkers. We apply this method to the Angenent
torus, the first known nontrivial example of a self-shrinker. We find that,
excluding dilations and translations, the index of the Angenent torus is $5$,
which is consistent with the lower bound of $3$ from the work of Liu and the
upper bound of $29$ from our earlier work. Also, we unexpectedly discover two
additional variations of the Angenent torus with eigenvalue $-1$.
