  We construct $13$ projective $\mathbb{Q}$-factorial Fano toric varieties and
show that for any minimal rational elliptic surface $X$ there is one such toric
variety $Z_X$ and a divisor class $\delta_X\in {\rm Cl}(Z_X)$ such that the
number of $(-1)$-curves of $X$ equals the dimension of the Riemann-Roch space
of $\delta_X$. As an application we give the number of $(-1)$-curves of any
such elliptic fibration of Halphen index $2$.
