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Let $E/\mathbb{Q}$ be an elliptic curve which has split multiplicativereduction at a prime $p$ and whose analytic rank $r_{an}(E)$ equals one. Themain goal of this article is to relate the second order derivative of theMazur-Tate-Teitelbaum $p$-adic $L$-function $L_p(E,s)$ of $E$ toNekov\'{a}\v{r}'s height pairing evaluated on natural elements arising from theBeilinson-Kato elements. Along the way, we extend a Rubin-style formula ofNekov\'a\v{r} (or in an alternative wording, correct another Rubin-styleformula of his) to apply in the presence of exceptional zeros. Our heightformula allows us, among other things, to compare the order of vanishing of$L_p(E,s)$ at $s=1$ to its (complex) analytic rank $r_{an}(E)$ assuming thenon-triviality of the height pairing. This has consequences towards aconjecture of Mazur, Tate and Teitelbaum.

On Nekov\'{a}\v{r}'s heights, exceptional zeros and a conjecture of Mazur-Tate-Teitelbaum