  We consider the focusing cubic wave equation in the energy supercritical
case, i.e., in dimensions $d \geq 5$. For this model an explicit nontrivial
self-similar blowup solution was recently found by the first and third author
in \cite{GlogicSchoerkhuber}. Furthermore, the solution is proven to be
co-dimension one stable in $d=7$. In this paper, we study the equation from a
numerical point of view. For $d=5$ and $d=7$ in the radial case, we provide
evidence that this solution is at the threshold between generic ODE blowup and
dispersion. In addition, we investigate the spectral problem that underlies the
stability analysis and compute the spectrum in general supercritical
dimensions.
