The aim of the paper is to prove the following result concerning moduli of curve families in the Heisenberg group. Let $\Omega$ be a domain in the Heisenberg group foliated by a family $\Gamma$ of legendrian curves. Assume that there is a quadratic differential $q$ on $\Omega$ in the kernel of an operator defined in \cite{Tim2} and every curve in $\Gamma$ is a horizontal trajectory for $q$. Let $l_\Gamma : \Omega \rightarrow ]0,+\infty[$ be the function that associates to a point $p\in \Omega$, the $q$-length of the leaf containing $p$. Then, the modulus of $\Gamma$ is \[ M_4 (\Gamma) = \int_\Omega \frac{|q|^2}{(l_\Gamma) ^4} \mathrm{d} L^3.\]