On the determinant bundles of abelian schemes

Let π : A → S be an abelian scheme over a scheme S which is quasi-projective over an affine noetherian scheme and let L be a symmetric, rigidified, relatively ample line bundle on A. We show that there is an isomorphism det(π∗L) ⊗24 (π∗ω∨A)⊗12d of line bundles on S, where d is the rank of the (locally free) sheaf π∗L. We also show that the numbers 24 and 12d are sharp in the following sense: if N > 1 is a common divisor of 12 and 24, then there are data as above such that det(π∗L) ⊗(24/N) (π∗ω∨A)⊗(12d/N).