We present a determination of the neutral current weak axial charge $G^Z_A(0)=-0.654(3)_{rm stat}(5)_{rm sys}$ using the strange quark axial charge $G^s_A(0)$ calculated with lattice QCD. We then perform a phenomenological analysis, where we combine the strange quark electromagnetic form factor from lattice QCD with (anti)neutrino-nucleon scattering differential cross section from MiniBooNE experiments in a momentum transfer region $0.24lesssim Q^2 lesssim 0.71$ GeV$^2$ to determine the neutral current weak axial form factor $G^Z_A(Q^2)$ in the range of $0lesssim Q^2leq 1$ GeV$^2$. This yields a phenomenological value of $G^Z_A(0)=-0.687(89)_{rm stat}(40)_{rm sys}$. The value of $G^Z_A(0)$ constrained by the lattice QCD calculation of $G^s_A(0)$, when compared to its phenomenological determination, provides a significant improvement in precision and accuracy and can be used to provide a constraint on the fit to $G^Z_A(Q^2)$ for $Q^2>0$. This constrained fit leads to an unambiguous determination of (anti)neutrino-nucleon neutral current elastic scattering differential cross section near $Q^2=0$ and can play an important role in numerically isolating nuclear effects in this region. We show a consistent description of $G^Z_A(Q^2)$ obtained from the (anti)neutrino-nucleon scattering cross section data requires a nonzero contribution of the strange quark electromagnetic form factor. We demonstrate the robustness of our analysis by providing a post-diction of the BNL E734 experimental data.
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