Let $t=\frac{1-\sqrt{5}}{2}$, then $$2t=1-\sqrt{5}\implies 2t-1=-\sqrt{5}\implies t^2 = t+1$$

It follows that $$t^4 = (t+1)^2 = t^2 + 2t + 1= 3t+2$$

And $$t^{12}=(3t+2)^3=(9t^2 + 12t + 4)(3t+2)=(21t+13)(3t+2)=63t^2+81t +26=144t + 189 =\boxed{161-72\sqrt{5}}$$