Number of points of bounded height of a certain singular cubic surface

By complex integration methods, we prove a very precise asymptotic formula about the number of rational points of height at most X on a certain toric variety. We estimate the cardinality V(X) := card{(x, y, z, t) is an element of (N boolean AND [1, X])(4) : (x, y, z, t) = 1 : xyz = t(3)}. Let N(X) := (log X)(3/5)(log(2) X)(-1/5). Then there exists a polynomial Q is an element of R[X] of degree 6 and a constant c > 0 such that, for X --> +infinity, we have V(X) = XQ(log X) + O(X1-1/8 exp{-cN(X)}). Futhermore, the leading coefficient of Q is 1/4 x 6! Pi p{(1 - 1/p)(7) (1 + 7/p + 1/p(2))}.