  We present here a formula for expressing the trace of the Frobenius
endomorphism of an elliptic curve $E$ over $\mathbb{F}_q$ satisfying $j(E)\neq
0,1728$ and $q\equiv 1 \pmod{12}$ in terms of special values of Gaussian
hypergeometric series. This paper uses methods introduced in Fuselier's work
for one parameter families of curves to express the trace of Frobenius of $E$
as a function of its $j$-invariant and discriminant instead of a parameter,
which are more intrinsic characteristics of the curve.
