Summary: We study periodic solutions for planar Newtonian \(N\)-body problem with equal masses, where each body is travelling along the same closed path. We provide a computer-assisted proof for the following facts: the local uniqueness and convexity of the Chenciner-Montgomery Eight [A. Chenciner and R. Montgomery, Ann. Math. (2) 152, No. 3, 881–901 (2000; Zbl 0987.70009)], the existence (and local uniqueness) for Gerver’s Super-Eight for 4 bodies, and a doubly symmetric linear chain for 6 bodies.