Let \(F_ N\) be the smooth curve defined over the rational numbers \(\mathbb{Q}\) by the equation \(x^ N+y^ N=1\). Then there is an exact sequence of localization \[ \coprod_{P\in F_ N(\overline\mathbb{Q})}K_ 2\mathbb{Q}(P)\to K_ 2F_ N\to K_ 2\mathbb{Q}(F_ N)@>\tau>> \coprod_{P\in F_ N(\overline\mathbb{Q})}\mathbb{Q}(P)^*. \] The author shows that the symbol \(\{1-xy,x\}^{6N}\in K_ 2\mathbb{Q}(F_ N)\) vanishes under \(\tau\), hence represents an element of \(K_ 2F_ N\). Furthermore he shows that the latter has non-zero image under the regulator homomorphism \(\hbox{reg}_{F_ N}:K_ 2F_ N\to H^ 1(F_ N(\mathbb{C})\), \(2\pi i\mathbb{R})^ +\) and that if \(N\geq 4\), the divisor of \(1-xy\) contains points which are of infnite order in \(\hbox{Jac}(F_ N)\). The above example is applied to show that if \(E\) is the CM elliptic curve with equation \(y^ 2=x^ 3+x\), then the symbol \(\{x-y,x\}^ 6\) represents an element of \(K_ 2 E\), which does not vanish under the regulator, which has non- torsion divisorial support, and which is in numerical agreement with part of the Beilinson conjectures.