  We lower bound the Faltings height of an abelian variety over a number field
by the sum of its injectivity diameter and the norm of its bad reduction
primes. It leads to an unconditional bound on the rank of Mordell-Weil groups.
Assuming the height conjecture of Lang and Silverman, we then obtain a
Northcott property for the regulator on the set of simple abelian varieties
defined over a fixed number field, of fixed dimension $g$, bounded rank and
with dense rational points over a number field. We remove the simplicity
assumption in the principally polarized case by giving a refined version of the
Lang-Silverman conjecture.
