Modular Virasoro vertex operator algebras with c=12

Using a $Z\left[\frac{1}{2}\right]$-form of Virasoro vertex operator algebra $L(\frac{1}{2},0)$ with central charge $\frac{1}{2}$, we obtain a modular vertex operator algebra over any field $F$ of finite characteristic different from 2. We determine the generators and classify the irreducible modules for this vertex operator algebra.