  We investigate exponential sums over singular binary quartic forms, proving
an explicit formula for the finite field Fourier transform of this set. Our
formula shares much in common with analogous formulas proved previously for
other vector spaces, but also exhibits a striking new feature: the point
counting function $a_p(E) = p + 1 - \#E(\mathbb{F}_p)$ associated to an
associated elliptic curve makes a prominent appearance. The proof techniques
are also new, involving techniques from elementary algebraic geometry and
classical invariant theory. As an application to prime number theory, we
demonstrate the existence of `many' 2-Selmer elements for elliptic curves with
discriminants that are squarefree and have at most four prime factors.
