In this paper, we study the bound of the number of isolated zeros of the Abelian integral $$I(h,\delta )$$ associated to system $$\dot{x}=y, \dot{y}=-x(x^2-1)(x^2+2)^2$$ under the perturbations $$\epsilon (\alpha _0+\alpha _1x^2+\alpha _2x^4+\alpha _3x^6)y \frac{\partial }{\partial y}$$ , where $$0< |\epsilon | \ll 1$$ and $$\alpha _i\in R$$ , $$i=0,1,2,3$$ . The period annulus of the unperturbed system is bounded by a two-saddle loop surrounding an elementary center. We divide the parameter space $$\{\alpha _0,\alpha _1,\alpha _2,\alpha _3 \}$$ into four parts, for each case the least upper bound of number of zeros of $$I(h,\delta )$$ will be investigated. We find that a smaller upper bound can be obtained when four generating elements of $$I(h,\delta )$$ are reduced to three special ones. Moreover, four zeros of $$I(h,\delta )$$ can be reached under case 4 by the asymptotic expansions of $$I(h,\delta )$$ and zero bifurcation theory.