In the literature there are several different definitions of what is often referred to as Beilinson's height pairing (see for example section 4.3.8 of Gillet and Soulé's paper [Arithmetic intersection theory][1], and this [note][2] of Müller-Stach). The height pairing is supposed to generalize the [Neron-Tate height pairing][3]. Is there a precise statement in the literature where Beilinson's height pairing for an elliptic curve over $\mathbb{Q}$ is compared to the Néron–Tate pairing? 


  [1]: http://www.numdam.org/item/PMIHES_1990__72__93_0/
  [2]: http://hodge.mathematik.uni-mainz.de/~stefan/papers/cime.pdf
  [3]: https://en.wikipedia.org/wiki/Néron–Tate_height