The Mori–Mukai determinant

I like Fano 3-folds. I like them so much that I created Fanography! One thing which I found intriguing in the story one aspect of the details of the Mori–Mukai classification that is not often mentioned. Namely, at the very end of their Classification of Fano threefolds with $\mathrm{b}_2\geq 2$, I, in paragraph (7.33), they introduce an invariant that they need to distinguish between various Fano 3-folds with otherwise the same numerical invariants. For instance, the families 2.22 and 2.24 have the same Hodge numbers and volume (the invariants that were most easily computed from their birational description), so, a priori, they could be the same family (or one a subfamily of the other). They needed a numerical deformation invariant to distinguish these families, and I found it frustrating that I could not find this invariant computed for all Fano 3-folds (Mori–Mukai only give it for the cases they need to distinguish), so that I could include it in Fanography. But this frustration is…