Given an elliptic curve over Q and a nontrivial element sigma of its Shafarevich--Tate group Sha(E), we introduce the *Visualization category* of abelian varieties that ``visualize'' sigma, in the sense of Cremona--Mazur, and we study minimal objects in this category, furnishing examples of their nonuniqueness. In particular, we show that there can be several minimal visualizing abelian varieties of different dimensions, answering a question of Mazur. This is joint work with Jerson Caro and Shiva Chidambaram.