[For part I see S. P. Cohen, ibid. 52, 427-444 (1986; Zbl 0567.14024).]
We study the arithmetic properties of torsion points on the commutative group variety given by a non-trivial additive extension of an elliptic curve defined over a number field, and show in particular that the Weil height of these points grows as the logarithm of their order, when this order is coprime to a finite set of exceptional primes. The results obtained are applied to derive explicit formulae for the p-adic periods, or Tate constants, of the height one and height two formal groups associated to elliptic curves with good reduction at the prime \(p\geq 5\), defined over a finite extension of the completion of \({\mathbb{Q}}\) at p.