The Cayley-Dickson order Lambda_p over O_{K_p}: what i^2=-1/p does and does not do

The deformation i² = −1/p (p > 0 squarefree) produces the imaginary quadratic field K_p = Q(√−p) and, by totally symmetric Cayley–Dickson doubling, a rank-8 lattice Λ_p. Classical facts, plus the index in a maximal order: For every such p, the Q-algebra O(−p,−p,−p) is a division algebra and is isomorphic to the unique division octonion algebra over Q (Cayley–Graves). The nine Heegner values of p do not yield nine pairwise non-isomorphic rational octonion algebras. The Gram matrix of Λ_p is g_{K_p} ⊗ diag(1,p) ⊗ diag(1,p), hence det T(Λ_p) = |D_{K_p}|^4 p^8. If O_max is a maximal order containing Λ_p, then [O_max : Λ_p] = |D_{K_p}|^2 p^4 (16 for p=1, Elkies–Gross; p^6 when p ≡ 3 mod 4). Hilbert symbols (p,−1)_v classify the quaternion algebra B_p, not an Arason class in H^3. The three quadratic axes of Λ_p are PIDs if and only if p is a Heegner number (Heegner–Baker–Stark). The nine cubes j(τ_p) = γ₂(τ_p)^3 and j(i/√2) = 8000 are due to Weber–Schertz.