The largest prime in the order of the Monster group is $71$. This number $71$ shows up at various places:

The minimal faithful representation has dimension $196883 = 47.59.71$

The Monster group can be realised as a Galois group $Gal$ $L(71)/{\mathbb{Q}(\sqrt{-71})}$ where $L(71)$ is a suitable field.

The appearance of $71$ in the above two cases (and possibly others) is not very surprising and might be reasoned out.

But the appearance of $71$ in a different area seems very intriguing:

The Monster is intimately connected to a special class of conformal field theories. These are the meromorphic $c = 24$ CFTs. The Monster here arises as the discrete automorphism group of the vertex operator algebra of one of the c =24 CFTs.

Schellekens in 1992 enumerated such CFTs and he found $71$ such CFTs! All these CFTs have a partition function of the form $$ Z(\tau) = j(\tau) + \mathcal{N} $$ where $j$ is modular invariant and $\mathcal{N} \geq -744$ is an integer. But any value of $\mathcal{N}$ won't work. Schellekens found $71$ values of $\mathcal{N}$ which will work.

Unfortunately, it is still not clear if the enumeration Schellekens made is exhaustive, i.e. if there are only exactly $71$ such theories.

Is the appearance of $71$ here just a coincidence? Or is it again connected to the Monster? It is hard to believe that this is just a coincidence.