regulator of an elliptic curve rational/irrational/transcendental?

Question

Let $K$ be a number field and $E/K$ an elliptic curve (or abelian variety) with $\mathrm{rk}\,E(K) > 0$. Can/will the elliptic (abelian) regulator $\mathrm{Reg}(E/K)$ be rational/irrational/transcendental?

Answer 1

There is currently no example of a number field $K$, an elliptic curve $E/K$, and a non-torsion point $P\in E(K)$, for which it is known that either $\hat h_E(P)$ or $\hat H_E(P)$ is not rational. However, there is an old result of Daniel Bertrand in which he shows in certain cases that the $p$-adic canonical height is transcendental over $\mathbb Q$, which is in fact how he proves that it's non-zero!