  We reexamine the disorder-dominated multicritical point of the
two-dimensional +/-J Ising model, known as the Nishimori point (NP). At the NP
we investigate numerically and analytically the behavior of the disorder
correlator, familiar from the self-dual description of the pure critical point
of the two-dimensional Ising model. We consider the logarithmic average and the
q-th moments of this correlator in the ensemble average over randomness, for
continuous q in the range 0<q<2.5, and demonstrate their conformal invariance.
At the NP we find, in contrast to the self-dual pure critical point, that the
disorder correlators exhibit multi-scaling in q which is different from that of
spin-spin correlators and that their scaling dimension becomes negative for q>1
and q<0. Using properties on the Nishimori line we show that the first moment
(q=1) of the disorder correlator is exactly one for all separations. The
spectrum of scaling dimensions at the NP is not parabolic in q.
