# 4 $v_1$-periodic classes

We finally get to the black classes, which are $v_1^4$-periodic. A “unit” of this $v_1$ periodicity looks like this:

The numbers on the class denote how many times it is $v_1^4$ divisible, relative to the coordinates in the diagram. For example, $x_{17, 8}$ can be divided by $v_1^4$ twice to give $x_{1, 0}$, while $x_{25, 4}$ doesn't even exist; only $v_1^4 x_{25, 4}$ does.

The divisibilities of unlabelled classes are determined by $h_0$ and $h_1$ products (if $x$ divides, then so do $h_0 x$ and $h_1 x$). If a class is completely unlabelled, then its label should be interpreted to be $0$.

Finally, the hollow classes are not actually in the diagram, but come from the C's. Their role is merely to indicate multiplications.

The differentials in $D$ follow from the differentials for ${\mathrm{tmf}}$ via the Leibniz rule again. They look as follows:

The two “hooks” with lower left corner at $(17, 8)$ and $(25, 4)$ are $v_1^4$ periodic. The classes left (including the hollow ones) are killed by elements in $k[v_1, w] \cdot x_{35, 6}$.