  Empirical analysis is often the first step towards the birth of a conjecture.
This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the
rational points on an elliptic curve, one of the most celebrated unsolved
problems in mathematics. Here we extend the original empirical approach, to the
analysis of the Cremona database of quantities relevant to BSD, inspecting more
than 2.5 million elliptic curves by means of the latest techniques in data
science, machine-learning and topological data analysis. Key quantities such as
rank, Weierstrass coefficients, period, conductor, Tamagawa number, regulator
and order of the Tate-Shafarevich group give rise to a high-dimensional
point-cloud whose statistical properties we investigate. We reveal patterns and
distributions in the rank versus Weierstrass coefficients, as well as the Beta
distribution of the BSD ratio of the quantities. Via gradient boosted trees,
machine learning is applied in finding inter-correlation amongst the various
quantities. We anticipate that our approach will spark further research on the
statistical properties of large datasets in Number Theory and more in general
in pure Mathematics.
