The tautological Chow ring of the moduli space \mathcal{A}_g of principally polarized abelian varieties of dimension g was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes from \mathcal{A}_g to the moduli space \mathcal{M}_g^{\mathrm{ct}} of genus g of curves of compact type, we prove that the product class [\mathcal{A}_1\times \mathcal{A}_5]\in \mathsf{CH}^{5}(\mathcal{A}_6) is non-tautological, the first construction of an interesting non-tautological algebraic class on the moduli spaces of abelian varieties. For our proof, we use the complete description of the the tautological ring \mathsf{R}^*(\mathcal{M}_6^{\mathrm{ct}}) in genus 6 conjectured by Pixton and recently proven by Canning-Larson-Schmitt. The tautological ring \mathsf{R}^*(\mathcal{M}_6^{\mathrm{ct}}) has a 1-dimensional Gorenstein kernel, which is geometrically explained by the Torelli pullback of [\mathcal{A}_1\times \mathcal{A}_5]. More generally, the Torelli pullback of the difference between [\mathcal{A}_1\times \mathcal{A}_{g-1}] and its tautological projection always lies in the Gorenstein kernel of \mathsf{R}^*(\mathcal{M}_g^{\mathrm{ct}}). The product map \mathcal{A}_1\times \mathcal{A}_{g-1}\rightarrow \mathcal{A}_g is a Noether-Lefschetz locus with general Neron-Severi rank 2. A natural extension of van der Geer's tautological ring is obtained by including more general Noether-Lefschetz loci. Results and conjectures related to cycle classes of Noether-Lefschetz loci for all g are presented.
