A few weeks ago Sergey Galkin explained a funny fact about the classification of Fano 3-folds of rank 1, due to Golyshev (the fact that is, not the classification, which is due to Mori–Mukai). Namely, if you make a table of these Fano 3-folds as follows,

1 2 3 4 5 6 7 8 9 11
1 $\mathrm{V}_2$ $\mathrm{V}_4$ $\mathrm{V}_6$ $\mathrm{V}_8$ $\mathrm{V}_{10}$ $\mathrm{V}_{12}$ $\mathrm{V}_{14}$ $\mathrm{V}_{16}$ $\mathrm{V}_{18}$ $\mathrm{V}_{22}$
2 $\mathrm{B}_1$ $\mathrm{B}_2$ $\mathrm{B}_3$ $\mathrm{B}_4$ $\mathrm{B}_5$
3 $Q$
4 $\mathbb{P}^3$

where the rows are indexed by the Fano index $r$, and the columns by the integer $N$, then

\begin{equation} I=r\psi(N) \end{equation}

where $\psi(N)=N\prod_{p\,\mid\, N}(1+1/p)$ is the Dedekind psi function, is a birational invariant. So whenever the value is 12 in the following table, the Fano variety is rational. You also see the correspondence between $\mathrm{B}_n$ and $\mathrm{V}_{4n+2}$.

1 2 3 4 5 6 7 8 9 11
1 1 3 4 6 6 12 8 12 12 12
2 2 6 8 12 12
3 12
4 12

At some point I'd like to do more with the classification of Fano 3-folds, I feel like something similar to superficie.info would work really well for this.

As a quick reminder, here are the descriptions of the varieties in the first table.

$\mathrm{V}_2$
sextic hypersurface in $\mathbb{P}(1,1,1,1,3)$
$\mathrm{V}_4$
quartic hypersurface in $\mathbb{P}^4$
$\mathrm{V}_6$
complete intersection of a quadric and cubic in $\mathbb{P}^5$
$\mathrm{V}_8$
complete intersection of 3 quadrics in $\mathbb{P}^6$
$\mathrm{V}_{10}$
section of $\mathrm{Gr}(2,5)$ by quadric and codimension 2 hyperplane plane
$\mathrm{V}_{12}$
section of $\mathrm{OGr}(5,10)$ by codimension 7 hyperplane
$\mathrm{V}_{14}$
section of $\mathrm{Gr}(2,6)$ by codimension 5 hyperplane
$\mathrm{V}_{16}$
section of $\mathrm{LGr}(3,6)$ by codimension 3 hyperplane
$\mathrm{V}_{18}$
section of $\mathrm{G}_2/P$ by codimension 2 hyperplane
$\mathrm{V}_{22}$
hard to describe succinctly
$\mathrm{B}_1$
sextic hypersurface in $\mathbb{P}(1,1,1,2,3)$
$\mathrm{B}_2$
quartic hypersurface in $\mathbb{P}(1,1,1,1,2)$
$\mathrm{B}_3$
cubic hypersurface in $\mathbb{P}^4$
$\mathrm{B}_4$
complete intersection of 2 quadrics in $\mathbb{P}^5$
$\mathrm{B}_5$
section of $\mathrm{Gr}(2,5)$ by codimension 3 hyperplane
$Q$
quadric hypersurface in $\mathbb{P}^4$