  We discuss the signatures of non-Gaussianity in the curvaton model where the
potential includes also a non-quadratic term. In such a case the non-linearity
parameter f_NL can become very small, and we show that non-Gaussianity is then
encoded in the non-reducible non-linearity parameter g_NL of the trispectrum,
which can be very large. Thus the place to look for the non-Gaussianity in the
curvaton model may be the trispectrum rather than the bispectrum. We also show
that g_NL measures directly the deviation of the curvaton potential from the
purely quadratic form. While g_NL depends on the strength of the non-quadratic
terms relative to the quadratic one, we find that for reasonable cases roughly
g_NL\sim O(-10^4)-O(-10^5), which are values that may well be accessible by
future observations.
