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We introduce a new computable invariant for strong shift equivalence ofshifts of finite type. The invariant is based on an invariant introduced byTrow, Boyle, and Marcus, but has the advantage of being readily computable.  We summarize briefly a large-scale numerical experiment aimed at decidingstrong shift equivalence for shifts of finite type given by irreducible$2\times 2$-matrices with entry sum less than 25, and give examplesillustrating to power of the new invariant, i.e., examples where the newinvariant can disprove strong shift equivalence whereas the other invariantsthat we use can not.

On some new invariants for strong shift equivalence for shifts of finite type