Arithmetic Enumeration of Klein-Graded Orders in a Biquadratic Field

Let K=Q(√3,√7) and O_K be its ring of integers. We classify the conductor ideals realized by the embedded Klein-graded orders R_(a,b,c)=Z⊕aZ√3⊕bZ√7⊕cZ√21, and determine the cardinality of every conductor fiber within this family. Every such conductor has the form 2t(√3)^(-ε_3 ) (√7)^(-ε_7 ) O_K, with ε_i∈{0,1} where ε_3=1 requires 3∣t and ε_7=1 requires 7∣t. Its fiber cardinality is a product of explicit local factors A(k)=1+3k(k+1)/2, B_0 (k)=(k+1)^2, and B_1 (k)=k(k+1)/2. We prove the classification by local conductor calculations, including a perfect-pairing argument at 2. Scalar conductor fibers, index spectra, and Dirichlet series by conductor and by index follow as consequences. The enumeration concerns embedded graded orders, not all orders or isomorphism classes of orders in K.