The arithmetic complexity of an elliptic curve defined over a number field is naturally quantified by the (stable) Faltings height. Faltings' spectrum is the set of all possible Faltings' heights. The corresponding spectrum for the Weil height on a projective space and the Neron-Tate height of an Abelian variety is dense on a semi-infinite interval. We show that, in contrast, Faltings' height has 2 isolated minima. We also determine the essential minimum of Faltings' height up to 5 decimal places. This is a joint work with Jose Burgos-Gil and Ricardo Menares.