  $q$-Supercongruences modulo the fifth and sixth powers of a cyclotomic
polynomial are very rare in the literature. In this paper, we establish some
$q$-supercongruences modulo the fifth and sixth powers of a cyclotomic
polynomial in terms of Jackson's $_8\phi_7$ summation, Watson's $_8\phi_7$
transformation, the creative microscoping method recently introduced by Guo and
Zudilin, and the Chinese remainder theorem for coprime polynomials. More
concretely, we give a $q$-analogue of a nice formula due to Long and
Ramakrishna [Adv. Math. 290 (2016), 773--808] and two $q$-supercongruences
involving double series.
