In this paper we prove that certain products and sums of powers of binomial coefficients modulo p = q f + 1 p = qf + 1 , q = a 2 + b 2 q = {a^2} + {b^2} , are determined by the parameters x occurring in distinct solutions of the quaternary quadratic partition \[ 16 p α = x 2 + 2 q u 2 + 2 q v 2 + q w 2 , ( x , u , v , w , p ) = 1 , x w = a v 2 − 2 b u v − a u 2 , x ≡ 4 ( mod q ) , α ⩾ 1. \begin {array}{*{20}{c}} {16{p^\alpha } = {x^2} + 2q{u^2} + 2q{v^2} + q{w^2},\quad (x,u,v,w,p) = 1,} \\ {xw = a{v^2} - 2buv - a{u^2},\quad x \equiv 4\pmod q,\alpha \geqslant 1.} \\ \end {array} \] The number of distinct solutions of this partition depends heavily on the class number of the imaginary cyclic quartic field \[ K = Q ( i 2 q + 2 a q ) , K = Q\left ( {i\sqrt {2q + 2a\sqrt q } } \right ), \] as well as on the number of roots of unity in K and on the way that p splits into prime ideals in the ring of integers of the field Q ( e 2 π i p / q ) Q({e^{2\pi ip/q}}) . Let the four cosets of the subgroup A of quartic residues be given by c j = 2 j A , j = 0 , 1 , 2 , 3 {c_j} = {2^j}A,j = 0,1,2,3 , and let \[ s j = 1 q ∑ t ∈ c j t , j = 0 , 1 , 2 , 3. {s_j} = \frac {1}{q}\sum \limits _{t \in {c_j}} {t,\quad j = 0,1,2,3.} \] Let s m {s_m} and s n {s_n} denote the smallest and next smallest of the s j {s_j} respectively. We give new, and unexpectedly simple determinations of Π k ∈ c n k f ! {\Pi _{k \in {c_n}}}kf! and Π k ∈ c n + 2 k f ! {\Pi _{k \in {c_{n + 2}}}}kf! , in terms of the parameters x in the above partition of 16 p α 16{p^\alpha } , in the complicated case that arises when the class number of K is > 1 > 1 and s m ≠ s n {s_m} \ne {s_n} .