AbstractLetAbe an abelian variety defined over a number fieldk, letpbe an odd prime number and let$F/k$be a cyclic extension ofp-power degree. Under not-too-stringent hypotheses we give an interpretation of thep-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points ofAof the${\rm Gal}(F/k)$-valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in situations in which thep-completion of the Mordell–Weil group ofAoverFis not a projective Galois module.