The local zeta function in enumerating quartic fields

An exact formula is obtained for the Fourier transform of the local condition of maximality modulo primes $p>3$ in the prehomogeneous vector space $2 \otimes $\mathrm{S}\mathrm{y}\mathrm{m}$^2($\mathbb{Z}$_p^3)$ parametrizing quartic fields, thus solving the local `quartic case’ in enumerating quartic fields.