Faltings elliptic curves in twisted Q-isogeny classes

Let G be the graph attached to the Q-isogeny class of an elliptic curve defined over Q: that is, a vertex for every elliptic curve defined over Q in the isogeny class, and edges in correspondence with the prime degree rational isogenies between them. Stevens shows that there is a unique elliptic curve in G with minimal Faltings height. We call this curve the Faltings elliptic curve in G. For every square-free integer d, we consider the graph~Gᵈ attached to the twisted elliptic curves in G by the quadratic character of Q(√d). It turns out that G and Gᵈ are canonically isomorphic as abstract graphs (the isomorphism identifies the vertices with equal j-invariant). In this paper we determine which vertex is the Faltings elliptic curve in Gᵈ. We also obtain the probability of a vertex in G to be the Faltings elliptic curve in~Gᵈ. It turns out that this probability depends on the p-adic valuations of rational values of certain modular functions.